module string | startPos dict | endPos dict | nextStartPos dict | goals list | goalsAfter list | ppTac string | elaborator string | kind string |
|---|---|---|---|---|---|---|---|---|
Mathlib.Order.Concept | {
"line": 418,
"column": 2
} | {
"line": 418,
"column": 95
} | {
"line": 419,
"column": 2
} | [
{
"pp": "α : Type u_2\nr' : α → α → Prop\nc' : Concept α α r'\ninst✝¹ : Std.Trichotomous r'\ninst✝ : IsTrans α r'\nx : α\nx✝ : x ∈ ⊤\nhx : x ∉ c'.extent\ny : α\nhy : y ∈ c'.extent\n⊢ r' y x",
"ppTerm": "?m.36",
"assigned": true,
"usedConstants": [
"Concept.mem_extent_of_rel_extent",
"Not... | [
"α : Type u_2\nr' : α → α → Prop\nc' : Concept α α r'\ninst✝¹ : Std.Trichotomous r'\ninst✝ : IsTrans α r'\nx : α\nx✝ : x ∈ ⊤\nhx : x ∉ c'.extent\ny : α\nhy : y ∈ c'.extent\n⊢ ¬x = y"
] | apply Not.imp_symm <| Std.Trichotomous.trichotomous x y (hx <| mem_extent_of_rel_extent · hy) | Lean.Elab.Tactic.evalApply | Lean.Parser.Tactic.apply |
Mathlib.Order.CompleteSublattice | {
"line": 104,
"column": 12
} | {
"line": 104,
"column": 22
} | {
"line": 104,
"column": 23
} | [
{
"pp": "α : Type u_1\ninst✝ : CompleteLattice α\nL : CompleteSublattice α\nι : Sort u_3\nf : ι → ↥L\n⊢ ↑(sSup (range f)) = ⨆ i, ↑(f i)",
"ppTerm": "?m.19",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"CompleteSublattice.instSupSet",
"congrArg",
"iSup",
"Membership.m... | [
"α : Type u_1\ninst✝ : CompleteLattice α\nL : CompleteSublattice α\nι : Sort u_3\nf : ι → ↥L\n⊢ ⨆ N ∈ range f, ↑N = ⨆ i, ↑(f i)"
] | coe_sSup', | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.Order.CompleteSublattice | {
"line": 152,
"column": 4
} | {
"line": 152,
"column": 62
} | {
"line": 152,
"column": 63
} | [
{
"pp": "α : Type u_1\nβ : Type u_2\ninst✝¹ : CompleteLattice α\ninst✝ : CompleteLattice β\nf : CompleteLatticeHom α β\nL✝ : CompleteSublattice α\nL : CompleteSublattice β\ns : Set α\nhs : s ⊆ ⇑f ⁻¹' ↑L\n⊢ sSup s ∈ ⇑f ⁻¹' ↑L",
"ppTerm": "?m.39",
"assigned": true,
"usedConstants": [
"Eq.mpr",
... | [
"α : Type u_1\nβ : Type u_2\ninst✝¹ : CompleteLattice α\ninst✝ : CompleteLattice β\nf : CompleteLatticeHom α β\nL✝ : CompleteSublattice α\nL : CompleteSublattice β\ns : Set α\nhs : s ⊆ ⇑f ⁻¹' ↑L\n⊢ sSup (⇑f '' s) ∈ L"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Order.CompleteSublattice | {
"line": 155,
"column": 4
} | {
"line": 155,
"column": 62
} | {
"line": 155,
"column": 63
} | [
{
"pp": "α : Type u_1\nβ : Type u_2\ninst✝¹ : CompleteLattice α\ninst✝ : CompleteLattice β\nf : CompleteLatticeHom α β\nL✝ : CompleteSublattice α\nL : CompleteSublattice β\ns : Set α\nhs : s ⊆ ⇑f ⁻¹' ↑L\n⊢ sInf s ∈ ⇑f ⁻¹' ↑L",
"ppTerm": "?m.57",
"assigned": true,
"usedConstants": [
"sInfHomCla... | [
"α : Type u_1\nβ : Type u_2\ninst✝¹ : CompleteLattice α\ninst✝ : CompleteLattice β\nf : CompleteLatticeHom α β\nL✝ : CompleteSublattice α\nL : CompleteSublattice β\ns : Set α\nhs : s ⊆ ⇑f ⁻¹' ↑L\n⊢ sInf (⇑f '' s) ∈ L"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Order.Concept | {
"line": 527,
"column": 2
} | {
"line": 527,
"column": 13
} | {
"line": 527,
"column": 14
} | [
{
"pp": "α : Type u_2\nβ : Type u_3\nr : α → β → Prop\nt : Set β\nc : Concept α β r\nh : c.intent ⊆ t\n⊢ c.intent ⊆ (ofAttributes r t).intent",
"ppTerm": "?m.17",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"congrArg",
"Concept.ofAttributes",
"Concept.intent",
"id",
... | [
"α : Type u_2\nβ : Type u_3\nr : α → β → Prop\nt : Set β\nc : Concept α β r\nh : c.intent ⊆ t\n⊢ c.intent ⊆ upperPolar r (lowerPolar r t)"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.SetTheory.Ordinal.Rank | {
"line": 48,
"column": 2
} | {
"line": 56,
"column": 42
} | {
"line": 58,
"column": 0
} | [
{
"pp": "α : Type u\na : α\nr : α → α → Prop\no : Ordinal.{u}\nha : Acc r a\nho : o ≤ ha.rank\n⊢ ∃ b, ∃ (hb : Acc r b), hb.rank = o",
"ppTerm": "?m.14",
"assigned": true,
"usedConstants": [
"Acc.rank_eq",
"LE.le.eq_or_lt",
"Ordinal.instLinearOrder",
"Preorder.toLT",
"Or... | [] | obtain rfl | ho := ho.eq_or_lt
· exact ⟨a, ha, rfl⟩
· revert ho
refine ha.recOn fun a ha IH ho ↦ ?_
rw [rank_eq, Ordinal.lt_iSup_iff] at ho
obtain ⟨⟨b, hb⟩, ho⟩ := ho
rw [Order.lt_succ_iff] at ho
obtain rfl | ho := ho.eq_or_lt
exacts [⟨b, ha b hb, rfl⟩, IH _ hb ho] | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.SetTheory.Ordinal.Rank | {
"line": 48,
"column": 2
} | {
"line": 56,
"column": 42
} | {
"line": 58,
"column": 0
} | [
{
"pp": "α : Type u\na : α\nr : α → α → Prop\no : Ordinal.{u}\nha : Acc r a\nho : o ≤ ha.rank\n⊢ ∃ b, ∃ (hb : Acc r b), hb.rank = o",
"ppTerm": "?m.14",
"assigned": true,
"usedConstants": [
"Acc.rank_eq",
"LE.le.eq_or_lt",
"Ordinal.instLinearOrder",
"Preorder.toLT",
"Or... | [] | obtain rfl | ho := ho.eq_or_lt
· exact ⟨a, ha, rfl⟩
· revert ho
refine ha.recOn fun a ha IH ho ↦ ?_
rw [rank_eq, Ordinal.lt_iSup_iff] at ho
obtain ⟨⟨b, hb⟩, ho⟩ := ho
rw [Order.lt_succ_iff] at ho
obtain rfl | ho := ho.eq_or_lt
exacts [⟨b, ha b hb, rfl⟩, IH _ hb ho] | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.Order.Filter.Partial | {
"line": 130,
"column": 4
} | {
"line": 131,
"column": 11
} | {
"line": 131,
"column": 12
} | [
{
"pp": "case mp\nα : Type u\nβ : Type v\nr : SetRel α β\nl₁ : Filter α\nl₂ : Filter β\n⊢ (∀ s ∈ l₂.sets, r.core s ∈ l₁) → l₁ ≤ rcomap r l₂",
"ppTerm": "?mp",
"assigned": true,
"usedConstants": [
"Filter.instMembership",
"Eq.mpr",
"Filter.mem_sets._simp_1",
"SetRel",
"c... | [
"case mp\nα : Type u\nβ : Type v\nr : SetRel α β\nl₁ : Filter α\nl₂ : Filter β\n⊢ (∀ s ∈ l₂, r.core s ∈ l₁) → ∀ (x : Set α), ∀ x_1 ∈ l₂, r.core x_1 ⊆ x → x ∈ l₁"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Order.Filter.Partial | {
"line": 132,
"column": 4
} | {
"line": 133,
"column": 11
} | {
"line": 133,
"column": 12
} | [
{
"pp": "case mpr\nα : Type u\nβ : Type v\nr : SetRel α β\nl₁ : Filter α\nl₂ : Filter β\n⊢ l₁ ≤ rcomap r l₂ → ∀ s ∈ l₂.sets, r.core s ∈ l₁",
"ppTerm": "?mpr",
"assigned": true,
"usedConstants": [
"Filter.instMembership",
"Eq.mpr",
"Filter.mem_sets._simp_1",
"SetRel",
"c... | [
"case mpr\nα : Type u\nβ : Type v\nr : SetRel α β\nl₁ : Filter α\nl₂ : Filter β\n⊢ (∀ (x : Set α), ∀ x_1 ∈ l₂, r.core x_1 ⊆ x → x ∈ l₁) → ∀ s ∈ l₂, r.core s ∈ l₁"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Order.Filter.Partial | {
"line": 163,
"column": 6
} | {
"line": 163,
"column": 57
} | {
"line": 164,
"column": 4
} | [
{
"pp": "case mp\nα : Type u\nβ : Type v\nγ : Type w\nr : SetRel α β\ns : SetRel β γ\nl : Filter γ\nt : Set α\nu : Set β\nh : r.preimage u ⊆ t\nv : Set γ\nvsets : v ∈ l\nhv : s.preimage v ⊆ u\n⊢ ∃ t_1 ∈ l, r.preimage (s.preimage t_1) ⊆ t",
"ppTerm": "?mp",
"assigned": true,
"usedConstants": [
... | [] | exact ⟨v, vsets, (SetRel.preimage_mono hv).trans h⟩ | Lean.Elab.Tactic.evalExact | Lean.Parser.Tactic.exact |
Mathlib.Order.Filter.Partial | {
"line": 181,
"column": 4
} | {
"line": 181,
"column": 42
} | {
"line": 181,
"column": 43
} | [
{
"pp": "case mp\nα : Type u\nβ : Type v\nr : SetRel α β\nl₁ : Filter α\nl₂ : Filter β\n⊢ l₁ ≤\n { sets := SetRel.image {(s, t) | r.preimage s ⊆ t} l₂.sets, univ_sets := ⋯, sets_of_superset := ⋯,\n inter_sets := ⋯ } →\n ∀ s ∈ l₂, r.preimage s ∈ l₁",
"ppTerm": "?mp",
"assigned": true,
... | [
"case mp\nα : Type u\nβ : Type v\nr : SetRel α β\nl₁ : Filter α\nl₂ : Filter β\n⊢ (∀ (x : Set α), ∀ x_1 ∈ l₂, r.preimage x_1 ⊆ x → x ∈ l₁) → ∀ s ∈ l₂, r.preimage s ∈ l₁"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Order.Filter.CardinalInter | {
"line": 236,
"column": 2
} | {
"line": 236,
"column": 31
} | {
"line": 236,
"column": 32
} | [
{
"pp": "ι α β : Type u\nc : Cardinal.{u}\nl✝ : Filter α\ninst✝¹ : CardinalInterFilter l✝ c\nl : Filter β\ninst✝ : CardinalInterFilter l c\nf : α → β\nS : Set (Set α)\nhSc : #↑S < c\nt : Set α → Set β\nhtl : ∀ s ∈ S, t s ∈ l\nht : ∀ s ∈ S, f ⁻¹' t s ⊆ s\n⊢ f ⁻¹' ⋂ i ∈ S, t i ⊆ ⋂₀ S",
"ppTerm": "?m.45",
... | [
"ι α β : Type u\nc : Cardinal.{u}\nl✝ : Filter α\ninst✝¹ : CardinalInterFilter l✝ c\nl : Filter β\ninst✝ : CardinalInterFilter l c\nf : α → β\nS : Set (Set α)\nhSc : #↑S < c\nt : Set α → Set β\nhtl : ∀ s ∈ S, t s ∈ l\nht : ∀ s ∈ S, f ⁻¹' t s ⊆ s\n⊢ ∀ t' ∈ S, ⋂ i ∈ S, f ⁻¹' t i ⊆ t'"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Order.Filter.Partial | {
"line": 182,
"column": 4
} | {
"line": 182,
"column": 42
} | {
"line": 182,
"column": 43
} | [
{
"pp": "case mpr\nα : Type u\nβ : Type v\nr : SetRel α β\nl₁ : Filter α\nl₂ : Filter β\n⊢ (∀ s ∈ l₂, r.preimage s ∈ l₁) →\n l₁ ≤\n { sets := SetRel.image {(s, t) | r.preimage s ⊆ t} l₂.sets, univ_sets := ⋯, sets_of_superset := ⋯,\n inter_sets := ⋯ }",
"ppTerm": "?mpr",
"assigned": true,
... | [
"case mpr\nα : Type u\nβ : Type v\nr : SetRel α β\nl₁ : Filter α\nl₂ : Filter β\n⊢ (∀ s ∈ l₂, r.preimage s ∈ l₁) → ∀ (x : Set α), ∀ x_1 ∈ l₂, r.preimage x_1 ⊆ x → x ∈ l₁"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Order.Filter.CardinalInter | {
"line": 313,
"column": 6
} | {
"line": 313,
"column": 31
} | {
"line": 313,
"column": 32
} | [
{
"pp": "case mp.basic\nα : Type u\nc : Cardinal.{u}\ng : Set (Set α)\ns✝ : Set α\nhreg : c.IsRegular\ns : Set α\nhs : s ∈ g\n⊢ #↑{s} < c ∧ ⋂₀ {s} ⊆ s",
"ppTerm": "?mp.basic",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"NonAssocSemiring.toAddCommMonoidWithOne",
"Set.fintypeSing... | [
"case mp.basic\nα : Type u\nc : Cardinal.{u}\ng : Set (Set α)\ns✝ : Set α\nhreg : c.IsRegular\ns : Set α\nhs : s ∈ g\n⊢ 1 < c"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Order.Height | {
"line": 69,
"column": 46
} | {
"line": 69,
"column": 95
} | {
"line": 69,
"column": 95
} | [
{
"pp": "α : Type u_1\ns : Set α\nr : α → α → Prop\nh : s.chainHeight r ≠ ⊤\nthis : Nonempty { t // t ⊆ s ∧ IsChain r t }\n⊢ ⨆ i, ?m.28 i < ⊤",
"ppTerm": "?m.30",
"assigned": true,
"usedConstants": [
"Set.chainHeight",
"Eq.mpr",
"Set.encard",
"Preorder.toLT",
"instCompl... | [] | by rwa [← chainHeight_eq_iSup, lt_top_iff_ne_top] | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.Order.Filter.CardinalInter | {
"line": 316,
"column": 25
} | {
"line": 316,
"column": 55
} | {
"line": 317,
"column": 4
} | [
{
"pp": "case mp.superset\nα : Type u\nc : Cardinal.{u}\ng : Set (Set α)\ns : Set α\nhreg : c.IsRegular\ns✝ t✝ : Set α\na✝¹ : CardinalGenerateSets g s✝\na✝ : s✝ ⊆ t✝\nih : ∃ S ⊆ g, #↑S < c ∧ ⋂₀ S ⊆ s✝\n⊢ ∃ S ⊆ g, #↑S < c ∧ ⋂₀ S ⊆ t✝",
"ppTerm": "?mp.superset",
"assigned": true,
"usedConstants": [
... | [] | exact Exists.imp (by tauto) ih | Lean.Elab.Tactic.evalExact | Lean.Parser.Tactic.exact |
Mathlib.Order.Filter.CardinalInter | {
"line": 316,
"column": 25
} | {
"line": 316,
"column": 55
} | {
"line": 317,
"column": 4
} | [
{
"pp": "case mp.superset\nα : Type u\nc : Cardinal.{u}\ng : Set (Set α)\ns : Set α\nhreg : c.IsRegular\ns✝ t✝ : Set α\na✝¹ : CardinalGenerateSets g s✝\na✝ : s✝ ⊆ t✝\nih : ∃ S ⊆ g, #↑S < c ∧ ⋂₀ S ⊆ s✝\n⊢ ∃ S ⊆ g, #↑S < c ∧ ⋂₀ S ⊆ t✝",
"ppTerm": "?mp.superset",
"assigned": true,
"usedConstants": [
... | [] | exact Exists.imp (by tauto) ih | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.Order.Filter.CardinalInter | {
"line": 316,
"column": 25
} | {
"line": 316,
"column": 55
} | {
"line": 317,
"column": 4
} | [
{
"pp": "case mp.superset\nα : Type u\nc : Cardinal.{u}\ng : Set (Set α)\ns : Set α\nhreg : c.IsRegular\ns✝ t✝ : Set α\na✝¹ : CardinalGenerateSets g s✝\na✝ : s✝ ⊆ t✝\nih : ∃ S ⊆ g, #↑S < c ∧ ⋂₀ S ⊆ s✝\n⊢ ∃ S ⊆ g, #↑S < c ∧ ⋂₀ S ⊆ t✝",
"ppTerm": "?mp.superset",
"assigned": true,
"usedConstants": [
... | [] | exact Exists.imp (by tauto) ih | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.Order.Height | {
"line": 109,
"column": 4
} | {
"line": 109,
"column": 15
} | {
"line": 109,
"column": 16
} | [
{
"pp": "case refine_1\nα : Type u_1\ns : Set α\nr : α → α → Prop\nh : ∀ a ⊆ s, IsChain r a → a = ∅\nx : α\n⊢ x ∈ s ↔ x ∈ ∅",
"ppTerm": "?refine_1",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"False",
"iff_false",
"Set.mem_empty_iff_false._simp_1",
"congrArg",
... | [
"case refine_1\nα : Type u_1\ns : Set α\nr : α → α → Prop\nh : ∀ a ⊆ s, IsChain r a → a = ∅\nx : α\n⊢ x ∉ s"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Order.Height | {
"line": 140,
"column": 29
} | {
"line": 140,
"column": 56
} | {
"line": 140,
"column": 57
} | [
{
"pp": "α : Type u_1\ns : Set α\nr : α → α → Prop\nn : ℕ\nhn : ↑n ≤ s.chainHeight (flip r)\na : Set α\nha₁ : a ⊆ s\nha₂ : a.encard = ↑n\nha₃ : IsChain (flip r) a\nx✝¹ : α\nhx : x✝¹ ∈ a\nx✝ : α\nhy : x✝ ∈ a\nhne : x✝¹ ≠ x✝\n⊢ (fun x y ↦ r x y ∨ r y x) x✝¹ x✝",
"ppTerm": "?m.58",
"assigned": true,
"u... | [
"α : Type u_1\ns : Set α\nr : α → α → Prop\nn : ℕ\nhn : ↑n ≤ s.chainHeight (flip r)\na : Set α\nha₁ : a ⊆ s\nha₂ : a.encard = ↑n\nha₃ : IsChain (flip r) a\nx✝¹ : α\nhx : x✝¹ ∈ a\nx✝ : α\nhy : x✝ ∈ a\nhne : x✝¹ ≠ x✝\n⊢ r x✝¹ x✝ ∨ r x✝ x✝¹"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Order.Height | {
"line": 140,
"column": 29
} | {
"line": 140,
"column": 56
} | {
"line": 140,
"column": 57
} | [
{
"pp": "α : Type u_1\ns : Set α\nr : α → α → Prop\nn : ℕ\nhn : ↑n ≤ s.chainHeight r\na : Set α\nha₁ : a ⊆ s\nha₂ : a.encard = ↑n\nha₃ : IsChain r a\nx✝¹ : α\nhx : x✝¹ ∈ a\nx✝ : α\nhy : x✝ ∈ a\nhne : x✝¹ ≠ x✝\n⊢ (fun x y ↦ flip r x y ∨ flip r y x) x✝¹ x✝",
"ppTerm": "?m.109",
"assigned": true,
"used... | [
"α : Type u_1\ns : Set α\nr : α → α → Prop\nn : ℕ\nhn : ↑n ≤ s.chainHeight r\na : Set α\nha₁ : a ⊆ s\nha₂ : a.encard = ↑n\nha₃ : IsChain r a\nx✝¹ : α\nhx : x✝¹ ∈ a\nx✝ : α\nhy : x✝ ∈ a\nhne : x✝¹ ≠ x✝\n⊢ r x✝¹ x✝ ∨ r x✝ x✝¹"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Order.Height | {
"line": 166,
"column": 2
} | {
"line": 166,
"column": 18
} | {
"line": 166,
"column": 19
} | [
{
"pp": "α : Type u_1\ns : Set α\nr : α → α → Prop\nhc :\n ((⇑(Subtype.relEmbedding (fun x1 x2 ↦ r x1 x2) fun x ↦ x ∈ s) '' univ).chainHeight fun x1 x2 ↦ r x1 x2) =\n univ.chainHeight (Subtype.val ⁻¹'o fun x1 x2 ↦ r x1 x2)\nhs : (Subtype.val ⁻¹'o fun x1 x2 ↦ r x1 x2) = fun x y ↦ r ↑x ↑y\n⊢ (univ.chainHeight... | [
"α : Type u_1\ns : Set α\nr : α → α → Prop\nhc :\n ((⇑(Subtype.relEmbedding (fun x1 x2 ↦ r x1 x2) fun x ↦ x ∈ s) '' univ).chainHeight fun x1 x2 ↦ r x1 x2) =\n univ.chainHeight (Subtype.val ⁻¹'o fun x1 x2 ↦ r x1 x2)\nhs : (Subtype.val ⁻¹'o fun x1 x2 ↦ r x1 x2) = fun x y ↦ r ↑x ↑y\n⊢ (univ.chainHeight fun x1 x2 ↦... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Order.Height | {
"line": 171,
"column": 2
} | {
"line": 171,
"column": 13
} | {
"line": 171,
"column": 14
} | [
{
"pp": "α : Type u_1\ns : Set α\ninst✝ : LE α\n⊢ (univ.chainHeight fun x1 x2 ↦ x1 ≤ x2) = s.chainHeight fun x1 x2 ↦ x1 ≤ x2",
"ppTerm": "?m.12",
"assigned": false,
"usedConstants": [],
"usedFVars": [],
"usedGoals": []
}
] | [
"α : Type u_1\ns : Set α\ninst✝ : LE α\n⊢ (univ.chainHeight fun x1 x2 ↦ x1 ≤ x2) = s.chainHeight fun x1 x2 ↦ x1 ≤ x2"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Order.Height | {
"line": 176,
"column": 2
} | {
"line": 176,
"column": 13
} | {
"line": 176,
"column": 14
} | [
{
"pp": "α : Type u_1\ns : Set α\ninst✝ : LT α\n⊢ (univ.chainHeight fun x1 x2 ↦ x1 < x2) = s.chainHeight fun x1 x2 ↦ x1 < x2",
"ppTerm": "?m.12",
"assigned": false,
"usedConstants": [],
"usedFVars": [],
"usedGoals": []
}
] | [
"α : Type u_1\ns : Set α\ninst✝ : LT α\n⊢ (univ.chainHeight fun x1 x2 ↦ x1 < x2) = s.chainHeight fun x1 x2 ↦ x1 < x2"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.NumberTheory.WellApproximable | {
"line": 331,
"column": 6
} | {
"line": 331,
"column": 42
} | {
"line": 331,
"column": 43
} | [
{
"pp": "case refine_1\nA : Type u_1\ninst✝⁵ : NormedAddCommGroup A\ninst✝⁴ : CompactSpace A\ninst✝³ : PreconnectedSpace A\ninst✝² : MeasurableSpace A\ninst✝¹ : BorelSpace A\nμ : Measure A\ninst✝ : μ.IsAddHaarMeasure\nξ : A\nn : ℕ\nhn : 0 < n\nδ : ℝ\nhδ : μ univ ≤ (n + 1) • μ (closedBall 0 (δ / 2))\nB : ↑(Icc 0... | [
"case refine_1\nA : Type u_1\ninst✝⁵ : NormedAddCommGroup A\ninst✝⁴ : CompactSpace A\ninst✝³ : PreconnectedSpace A\ninst✝² : MeasurableSpace A\ninst✝¹ : BorelSpace A\nμ : Measure A\ninst✝ : μ.IsAddHaarMeasure\nξ : A\nn : ℕ\nhn : 0 < n\nδ : ℝ\nhδ : μ univ ≤ (n + 1) • μ (closedBall 0 (δ / 2))\nB : ↑(Icc 0 n) → Set A ... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Order.Interval.Set.Nat | {
"line": 23,
"column": 2
} | {
"line": 23,
"column": 38
} | {
"line": 23,
"column": 39
} | [
{
"pp": "a b : ℕ\n⊢ (Icc a b).ncard = b + 1 - a",
"ppTerm": "?m.15",
"assigned": false,
"usedConstants": [],
"usedFVars": [],
"usedGoals": []
}
] | [
"a b : ℕ\n⊢ (Icc a b).ncard = b + 1 - a"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Order.Interval.Set.Nat | {
"line": 26,
"column": 2
} | {
"line": 26,
"column": 38
} | {
"line": 26,
"column": 39
} | [
{
"pp": "a b : ℕ\n⊢ (Ico a b).ncard = b - a",
"ppTerm": "?m.9",
"assigned": false,
"usedConstants": [],
"usedFVars": [],
"usedGoals": []
}
] | [
"a b : ℕ\n⊢ (Ico a b).ncard = b - a"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Order.Interval.Set.Nat | {
"line": 29,
"column": 2
} | {
"line": 29,
"column": 38
} | {
"line": 29,
"column": 39
} | [
{
"pp": "a b : ℕ\n⊢ (Ioc a b).ncard = b - a",
"ppTerm": "?m.9",
"assigned": false,
"usedConstants": [],
"usedFVars": [],
"usedGoals": []
}
] | [
"a b : ℕ\n⊢ (Ioc a b).ncard = b - a"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Order.Interval.Set.Nat | {
"line": 32,
"column": 2
} | {
"line": 32,
"column": 38
} | {
"line": 32,
"column": 39
} | [
{
"pp": "a b : ℕ\n⊢ (Ioo a b).ncard = b - a - 1",
"ppTerm": "?m.15",
"assigned": false,
"usedConstants": [],
"usedFVars": [],
"usedGoals": []
}
] | [
"a b : ℕ\n⊢ (Ioo a b).ncard = b - a - 1"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Order.Interval.Set.Nat | {
"line": 35,
"column": 2
} | {
"line": 35,
"column": 38
} | {
"line": 35,
"column": 39
} | [
{
"pp": "a b : ℕ\n⊢ (uIcc a b).ncard = (↑b - ↑a).natAbs + 1",
"ppTerm": "?m.15",
"assigned": false,
"usedConstants": [],
"usedFVars": [],
"usedGoals": []
}
] | [
"a b : ℕ\n⊢ (uIcc a b).ncard = (↑b - ↑a).natAbs + 1"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Order.Interval.Set.Nat | {
"line": 38,
"column": 2
} | {
"line": 38,
"column": 38
} | {
"line": 38,
"column": 39
} | [
{
"pp": "b : ℕ\n⊢ (Iic b).ncard = b + 1",
"ppTerm": "?m.11",
"assigned": false,
"usedConstants": [],
"usedFVars": [],
"usedGoals": []
}
] | [
"b : ℕ\n⊢ (Iic b).ncard = b + 1"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Order.Interval.Set.Nat | {
"line": 41,
"column": 2
} | {
"line": 41,
"column": 38
} | {
"line": 41,
"column": 39
} | [
{
"pp": "b : ℕ\n⊢ (Iio b).ncard = b",
"ppTerm": "?m.5",
"assigned": false,
"usedConstants": [],
"usedFVars": [],
"usedGoals": []
}
] | [
"b : ℕ\n⊢ (Iio b).ncard = b"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Order.Filter.Cocardinal | {
"line": 100,
"column": 2
} | {
"line": 100,
"column": 38
} | {
"line": 100,
"column": 39
} | [
{
"pp": "α : Type u\nc : Cardinal.{u}\nhreg : c.IsRegular\nx : α\n⊢ ∀ᶠ (a : α) in cocardinal α hreg, a ≠ x",
"ppTerm": "?m.5",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"NonAssocSemiring.toAddCommMonoidWithOne",
"Set.fintypeSingleton",
"Preorder.toLT",
"Classical.n... | [
"α : Type u\nc : Cardinal.{u}\nhreg : c.IsRegular\nx : α\n⊢ 1 < c"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.NumberTheory.WellApproximable | {
"line": 351,
"column": 2
} | {
"line": 351,
"column": 13
} | {
"line": 351,
"column": 14
} | [
{
"pp": "A : Type u_1\ninst✝⁵ : NormedAddCommGroup A\ninst✝⁴ : CompactSpace A\ninst✝³ : PreconnectedSpace A\ninst✝² : MeasurableSpace A\ninst✝¹ : BorelSpace A\nμ : Measure A\ninst✝ : μ.IsAddHaarMeasure\nξ : A\nn : ℕ\nhn : 0 < n\nδ : ℝ\nB : ↑(Icc 0 n) → Set A := fun j ↦ closedBall (↑j • ξ) (δ / 2)\nhB : ∀ (j : ↑... | [
"A : Type u_1\ninst✝⁵ : NormedAddCommGroup A\ninst✝⁴ : CompactSpace A\ninst✝³ : PreconnectedSpace A\ninst✝² : MeasurableSpace A\ninst✝¹ : BorelSpace A\nμ : Measure A\ninst✝ : μ.IsAddHaarMeasure\nξ : A\nn : ℕ\nhn : 0 < n\nδ : ℝ\nB : ↑(Icc 0 n) → Set A := fun j ↦ closedBall (↑j • ξ) (δ / 2)\nhB : ∀ (j : ↑(Icc 0 n)), ... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Order.KonigLemma | {
"line": 69,
"column": 46
} | {
"line": 69,
"column": 57
} | {
"line": 69,
"column": 58
} | [
{
"pp": "α : Type u_1\ninst✝¹ : PartialOrder α\ninst✝ : IsStronglyAtomic α\nb : α\nhfin : ∀ (a : α), {x | a ⋖ x}.Finite\nhb : (Ici b).Infinite\nh : ∀ (a : { a // (Ici a).Infinite }), ∃ b, ↑a ⋖ b ∧ (Ici b).Infinite :=\n fun a ↦ exists_covby_infinite_Ici_of_infinite_Ici a.property (hfin ↑a)\nks : ℕ → { a // (Ici... | [
"α : Type u_1\ninst✝¹ : PartialOrder α\ninst✝ : IsStronglyAtomic α\nb : α\nhfin : ∀ (a : α), {x | a ⋖ x}.Finite\nhb : (Ici b).Infinite\nh : ∀ (a : { a // (Ici a).Infinite }), ∃ b, ↑a ⋖ b ∧ (Ici b).Infinite :=\n fun a ↦ exists_covby_infinite_Ici_of_infinite_Ici a.property (hfin ↑a)\nks : ℕ → { a // (Ici a).Infinite... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Order.KonigLemma | {
"line": 80,
"column": 62
} | {
"line": 80,
"column": 73
} | {
"line": 80,
"column": 74
} | [
{
"pp": "α : Type u_1\ninst✝³ : PartialOrder α\ninst✝² : IsStronglyAtomic α\ninst✝¹ : OrderBot α\ninst✝ : Infinite α\nhfin : ∀ (a : α), {x | a ⋖ x}.Finite\n⊢ (Ici ⊥).Infinite",
"ppTerm": "?m.34",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"Set.Ici",
"congrArg",
"Set.univ"... | [
"α : Type u_1\ninst✝³ : PartialOrder α\ninst✝² : IsStronglyAtomic α\ninst✝¹ : OrderBot α\ninst✝ : Infinite α\nhfin : ∀ (a : α), {x | a ⋖ x}.Finite\n⊢ univ.Infinite"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Order.KonigLemma | {
"line": 89,
"column": 17
} | {
"line": 89,
"column": 70
} | {
"line": 89,
"column": 71
} | [
{
"pp": "case succ\nα : Type u_1\ninst✝⁴ : PartialOrder α\ninst✝³ : IsStronglyAtomic α\ninst✝² : GradeMinOrder ℕ α\ninst✝¹ : OrderBot α\ninst✝ : Infinite α\nhfin : ∀ (a : α), {x | a ⋖ x}.Finite\nf : ℕ ↪o α\nh0 : f 0 = ⊥\nhf : ∀ (i : ℕ), f i ⋖ f (i + 1)\ni : ℕ\nih : grade ℕ (f i) = i\n⊢ grade ℕ (f (i + 1)) = i +... | [
"case succ\nα : Type u_1\ninst✝⁴ : PartialOrder α\ninst✝³ : IsStronglyAtomic α\ninst✝² : GradeMinOrder ℕ α\ninst✝¹ : OrderBot α\ninst✝ : Infinite α\nhfin : ∀ (a : α), {x | a ⋖ x}.Finite\nf : ℕ ↪o α\nh0 : f 0 = ⊥\nhf : ∀ (i : ℕ), f i ⋖ f (i + 1)\ni : ℕ\nih : grade ℕ (f i) = i\n⊢ i + 1 = grade ℕ (f (i + 1))"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Order.Lattice.Congruence | {
"line": 49,
"column": 22
} | {
"line": 49,
"column": 33
} | {
"line": 49,
"column": 34
} | [
{
"pp": "α : Type u_2\ninst✝ : Lattice α\nc : LatticeCon α\nx y : α\nh : c.toSetoid (x ⊓ y) (x ⊔ y)\n⊢ x ≈ ?m.17 h",
"ppTerm": "?m.15",
"assigned": false,
"usedConstants": [],
"usedFVars": [],
"usedGoals": []
}
] | [
"α : Type u_2\ninst✝ : Lattice α\nc : LatticeCon α\nx y : α\nh : c.toSetoid (x ⊓ y) (x ⊔ y)\n⊢ x ≈ ?m.17 h"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Order.Lattice.Congruence | {
"line": 49,
"column": 67
} | {
"line": 49,
"column": 78
} | {
"line": 49,
"column": 79
} | [
{
"pp": "α : Type u_2\ninst✝ : Lattice α\nc : LatticeCon α\nx y : α\nh : c.toSetoid (x ⊓ y) (x ⊔ y)\n⊢ x ⊓ y ≈ y",
"ppTerm": "?m.16",
"assigned": false,
"usedConstants": [],
"usedFVars": [],
"usedGoals": []
}
] | [
"α : Type u_2\ninst✝ : Lattice α\nc : LatticeCon α\nx y : α\nh : c.toSetoid (x ⊓ y) (x ⊔ y)\n⊢ x ⊓ y ≈ y"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Order.Lattice.Congruence | {
"line": 50,
"column": 23
} | {
"line": 50,
"column": 34
} | {
"line": 50,
"column": 35
} | [
{
"pp": "α : Type u_2\ninst✝ : Lattice α\nc : LatticeCon α\nx y : α\nh : c.toSetoid x y\n⊢ x ⊓ y ≈ ?m.27 h",
"ppTerm": "?m.25",
"assigned": false,
"usedConstants": [],
"usedFVars": [],
"usedGoals": []
}
] | [
"α : Type u_2\ninst✝ : Lattice α\nc : LatticeCon α\nx y : α\nh : c.toSetoid x y\n⊢ x ⊓ y ≈ ?m.27 h"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Order.Lattice.Congruence | {
"line": 50,
"column": 59
} | {
"line": 50,
"column": 70
} | {
"line": 50,
"column": 71
} | [
{
"pp": "α : Type u_2\ninst✝ : Lattice α\nc : LatticeCon α\nx y : α\nh : c.toSetoid x y\n⊢ y ≈ x ⊔ y",
"ppTerm": "?m.26",
"assigned": false,
"usedConstants": [],
"usedFVars": [],
"usedGoals": []
}
] | [
"α : Type u_2\ninst✝ : Lattice α\nc : LatticeCon α\nx y : α\nh : c.toSetoid x y\n⊢ y ≈ x ⊔ y"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Order.Lattice.Congruence | {
"line": 58,
"column": 4
} | {
"line": 58,
"column": 55
} | {
"line": 58,
"column": 56
} | [
{
"pp": "α : Type u_2\ninst✝ : Lattice α\nr : α → α → Prop\nh₂ : ∀ ⦃x y : α⦄, r x y ↔ r (x ⊓ y) (x ⊔ y)\nh₄ : ∀ ⦃x y t : α⦄, x ≤ y → r x y → r (x ⊓ t) (y ⊓ t) ∧ r (x ⊔ t) (y ⊔ t)\na b c d : α\nhab : a ≤ b\nhbd : b ≤ d\nhac : a ≤ c\nhcd : c ≤ d\nhad : r a d\nthis : r (b ⊓ c ⊓ (b ⊔ c)) (d ⊓ (b ⊔ c))\n⊢ r b c",
... | [
"α : Type u_2\ninst✝ : Lattice α\nr : α → α → Prop\nh₂ : ∀ ⦃x y : α⦄, r x y ↔ r (x ⊓ y) (x ⊔ y)\nh₄ : ∀ ⦃x y t : α⦄, x ≤ y → r x y → r (x ⊓ t) (y ⊓ t) ∧ r (x ⊔ t) (y ⊔ t)\na b c d : α\nhab : a ≤ b\nhbd : b ≤ d\nhac : a ≤ c\nhcd : c ≤ d\nhad : r a d\nthis : r (b ⊓ c ⊓ (b ⊔ c)) (d ⊓ (b ⊔ c))\n⊢ r (b ⊓ c ⊓ (b ⊔ c)) (b... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Order.Lattice.Congruence | {
"line": 60,
"column": 2
} | {
"line": 60,
"column": 90
} | {
"line": 61,
"column": 4
} | [
{
"pp": "α : Type u_2\ninst✝ : Lattice α\nr : α → α → Prop\nh₂ : ∀ ⦃x y : α⦄, r x y ↔ r (x ⊓ y) (x ⊔ y)\nh₄ : ∀ ⦃x y t : α⦄, x ≤ y → r x y → r (x ⊓ t) (y ⊓ t) ∧ r (x ⊔ t) (y ⊔ t)\na b c d : α\nhab : a ≤ b\nhbd : b ≤ d\nhac : a ≤ c\nhcd : c ≤ d\nhad : r a d\n⊢ r (b ⊓ c) d",
"ppTerm": "?m.55",
"assigned":... | [
"α : Type u_2\ninst✝ : Lattice α\nr : α → α → Prop\nh₂ : ∀ ⦃x y : α⦄, r x y ↔ r (x ⊓ y) (x ⊔ y)\nh₄ : ∀ ⦃x y t : α⦄, x ≤ y → r x y → r (x ⊓ t) (y ⊓ t) ∧ r (x ⊔ t) (y ⊔ t)\na b c d : α\nhab : a ≤ b\nhbd : b ≤ d\nhac : a ≤ c\nhcd : c ≤ d\nhad : r a d\n⊢ r (b ⊓ c) d"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Order.Lattice.Congruence | {
"line": 72,
"column": 12
} | {
"line": 72,
"column": 35
} | {
"line": 72,
"column": 36
} | [
{
"pp": "α : Type u_2\ninst✝ : Lattice α\nr : α → α → Prop\nh₂ : ∀ ⦃x y : α⦄, r x y ↔ r (x ⊓ y) (x ⊔ y)\nh₃ : ∀ ⦃x y z : α⦄, x ≤ y → y ≤ z → r x y → r y z → r x z\nh₄ : ∀ ⦃x y t : α⦄, x ≤ y → r x y → r (x ⊓ t) (y ⊓ t) ∧ r (x ⊔ t) (y ⊔ t)\nx y z : α\nhxy : r x y\nhyz : r y z\n⊢ x ⊓ y ⊓ z ≤ y ⊔ z",
"ppTerm": ... | [
"α : Type u_2\ninst✝ : Lattice α\nr : α → α → Prop\nh₂ : ∀ ⦃x y : α⦄, r x y ↔ r (x ⊓ y) (x ⊔ y)\nh₃ : ∀ ⦃x y z : α⦄, x ≤ y → y ≤ z → r x y → r y z → r x z\nh₄ : ∀ ⦃x y t : α⦄, x ≤ y → r x y → r (x ⊓ t) (y ⊓ t) ∧ r (x ⊔ t) (y ⊔ t)\nx y z : α\nhxy : r x y\nhyz : r y z\n⊢ x ⊓ (y ⊓ z) ≤ y ⊔ z"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Order.Lattice.Congruence | {
"line": 74,
"column": 8
} | {
"line": 74,
"column": 31
} | {
"line": 74,
"column": 32
} | [
{
"pp": "α : Type u_2\ninst✝ : Lattice α\nr : α → α → Prop\nh₂ : ∀ ⦃x y : α⦄, r x y ↔ r (x ⊓ y) (x ⊔ y)\nh₃ : ∀ ⦃x y z : α⦄, x ≤ y → y ≤ z → r x y → r y z → r x z\nh₄ : ∀ ⦃x y t : α⦄, x ≤ y → r x y → r (x ⊓ t) (y ⊓ t) ∧ r (x ⊔ t) (y ⊔ t)\nx y z : α\nhxy : r x y\nhyz : r y z\n⊢ x ⊓ y ⊓ z ≤ y ⊓ z",
"ppTerm": ... | [
"α : Type u_2\ninst✝ : Lattice α\nr : α → α → Prop\nh₂ : ∀ ⦃x y : α⦄, r x y ↔ r (x ⊓ y) (x ⊔ y)\nh₃ : ∀ ⦃x y z : α⦄, x ≤ y → y ≤ z → r x y → r y z → r x z\nh₄ : ∀ ⦃x y t : α⦄, x ≤ y → r x y → r (x ⊓ t) (y ⊓ t) ∧ r (x ⊔ t) (y ⊔ t)\nx y z : α\nhxy : r x y\nhyz : r y z\n⊢ x ⊓ (y ⊓ z) ≤ y ∧ x ⊓ (y ⊓ z) ≤ z"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Order.Lattice.Congruence | {
"line": 77,
"column": 8
} | {
"line": 77,
"column": 56
} | {
"line": 77,
"column": 57
} | [
{
"pp": "α : Type u_2\ninst✝ : Lattice α\nr : α → α → Prop\nh₂ : ∀ ⦃x y : α⦄, r x y ↔ r (x ⊓ y) (x ⊔ y)\nh₃ : ∀ ⦃x y z : α⦄, x ≤ y → y ≤ z → r x y → r y z → r x z\nh₄ : ∀ ⦃x y t : α⦄, x ≤ y → r x y → r (x ⊓ t) (y ⊓ t) ∧ r (x ⊔ t) (y ⊔ t)\nx y z : α\nhxy : r x y\nhyz : r y z\nthis : r (x ⊓ y ⊓ (y ⊓ z)) ((x ⊔ y) ... | [
"α : Type u_2\ninst✝ : Lattice α\nr : α → α → Prop\nh₂ : ∀ ⦃x y : α⦄, r x y ↔ r (x ⊓ y) (x ⊔ y)\nh₃ : ∀ ⦃x y z : α⦄, x ≤ y → y ≤ z → r x y → r y z → r x z\nh₄ : ∀ ⦃x y t : α⦄, x ≤ y → r x y → r (x ⊓ t) (y ⊓ t) ∧ r (x ⊔ t) (y ⊔ t)\nx y z : α\nhxy : r x y\nhyz : r y z\nthis : r (x ⊓ y ⊓ (y ⊓ z)) ((x ⊔ y) ⊓ (y ⊓ z))\n... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Order.Nucleus | {
"line": 216,
"column": 49
} | {
"line": 216,
"column": 65
} | {
"line": 216,
"column": 66
} | [
{
"pp": "X : Type u_1\ninst✝ : Frame X\nn✝ m✝ : Nucleus X\nx✝ y✝ : X\nm n : Nucleus X\nx y : X\nthis : Nonempty X\nk : X\nhxyk : k ≥ x ⊓ y\nl : X\nhlx : ∀ x_1 ≥ x, l ⊓ m x_1 ≤ n x_1\nhly : ∀ i ≥ y, l ⊓ m i ≤ n i\nhlk : l ≤ m k\n⊢ l = l ⊓ m ((x ⊔ k) ⊓ (y ⊔ k))",
"ppTerm": "?m.212",
"assigned": true,
... | [
"X : Type u_1\ninst✝ : Frame X\nn✝ m✝ : Nucleus X\nx✝ y✝ : X\nm n : Nucleus X\nx y : X\nthis : Nonempty X\nk : X\nhxyk : k ≥ x ⊓ y\nl : X\nhlx : ∀ x_1 ≥ x, l ⊓ m x_1 ≤ n x_1\nhly : ∀ i ≥ y, l ⊓ m i ≤ n i\nhlk : l ≤ m k\n⊢ l = l ⊓ m (x ⊓ y ⊔ k)"
] | ← sup_inf_right, | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.Order.Lattice.Congruence | {
"line": 79,
"column": 8
} | {
"line": 79,
"column": 73
} | {
"line": 80,
"column": 10
} | [
{
"pp": "α : Type u_2\ninst✝ : Lattice α\nr : α → α → Prop\nh₂ : ∀ ⦃x y : α⦄, r x y ↔ r (x ⊓ y) (x ⊔ y)\nh₃ : ∀ ⦃x y z : α⦄, x ≤ y → y ≤ z → r x y → r y z → r x z\nh₄ : ∀ ⦃x y t : α⦄, x ≤ y → r x y → r (x ⊓ t) (y ⊓ t) ∧ r (x ⊔ t) (y ⊔ t)\nx y z : α\nhxy : r x y\nhyz : r y z\n⊢ r (y ⊔ z) (x ⊔ y ⊔ z)",
"ppTer... | [
"α : Type u_2\ninst✝ : Lattice α\nr : α → α → Prop\nh₂ : ∀ ⦃x y : α⦄, r x y ↔ r (x ⊓ y) (x ⊔ y)\nh₃ : ∀ ⦃x y z : α⦄, x ≤ y → y ≤ z → r x y → r y z → r x z\nh₄ : ∀ ⦃x y t : α⦄, x ≤ y → r x y → r (x ⊓ t) (y ⊓ t) ∧ r (x ⊔ t) (y ⊔ t)\nx y z : α\nhxy : r x y\nhyz : r y z\n⊢ r (y ⊔ z) (x ⊔ (y ⊔ z))"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Order.KonigLemma | {
"line": 138,
"column": 36
} | {
"line": 138,
"column": 57
} | {
"line": 138,
"column": 57
} | [
{
"pp": "α : ℕ → Type u_1\ninst✝¹ : Finite (α 0)\ninst✝ : ∀ (i : ℕ), Nonempty (α i)\nπ : {i j : ℕ} → i ≤ j → α j → α i\nπ_refl : ∀ ⦃i : ℕ⦄ (a : α i), π ⋯ a = a\nπ_trans : ∀ ⦃i j k : ℕ⦄ (hij : i ≤ j) (hjk : j ≤ k) (a : α k), π hij (π hjk a) = π ⋯ a\nhfin : ∀ (i : ℕ) (a : α i), {b | π ⋯ b = a}.Finite\nαs : Type u... | [] | by rw [π_trans, ← h2] | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.Order.Lattice.Congruence | {
"line": 91,
"column": 21
} | {
"line": 91,
"column": 63
} | {
"line": 91,
"column": 64
} | [
{
"pp": "F : Type u_1\nα : Type u_2\nβ : Type u_3\ninst✝¹ : Lattice α\ninst✝ : Lattice β\nr : α → α → Prop\nh₁ : Std.Refl r\nh₂ : ∀ ⦃x y : α⦄, r x y ↔ r (x ⊓ y) (x ⊔ y)\nh₃ : ∀ ⦃x y z : α⦄, x ≤ y → y ≤ z → r x y → r y z → r x z\nh₄ : ∀ ⦃x y t : α⦄, x ≤ y → r x y → r (x ⊓ t) (y ⊓ t) ∧ r (x ⊔ t) (y ⊔ t)\nx✝ y✝ : ... | [
"F : Type u_1\nα : Type u_2\nβ : Type u_3\ninst✝¹ : Lattice α\ninst✝ : Lattice β\nr : α → α → Prop\nh₁ : Std.Refl r\nh₂ : ∀ ⦃x y : α⦄, r x y ↔ r (x ⊓ y) (x ⊔ y)\nh₃ : ∀ ⦃x y z : α⦄, x ≤ y → y ≤ z → r x y → r y z → r x z\nh₄ : ∀ ⦃x y t : α⦄, x ≤ y → r x y → r (x ⊓ t) (y ⊓ t) ∧ r (x ⊔ t) (y ⊔ t)\nx✝ y✝ : α\nh : r x✝ ... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Order.Nucleus | {
"line": 220,
"column": 37
} | {
"line": 220,
"column": 53
} | {
"line": 220,
"column": 54
} | [
{
"pp": "X : Type u_1\ninst✝ : Frame X\nn✝ m✝ : Nucleus X\nx✝ y✝ : X\nm n : Nucleus X\nx y : X\nthis : Nonempty X\nk : X\nhxyk : k ≥ x ⊓ y\nl : X\nhlx : ∀ x_1 ≥ x, l ⊓ m x_1 ≤ n x_1\nhly : ∀ i ≥ y, l ⊓ m i ≤ n i\nhlk : l ≤ m k\n⊢ n ((x ⊔ k) ⊓ (y ⊔ k)) = n k",
"ppTerm": "?m.258",
"assigned": true,
"u... | [
"X : Type u_1\ninst✝ : Frame X\nn✝ m✝ : Nucleus X\nx✝ y✝ : X\nm n : Nucleus X\nx y : X\nthis : Nonempty X\nk : X\nhxyk : k ≥ x ⊓ y\nl : X\nhlx : ∀ x_1 ≥ x, l ⊓ m x_1 ≤ n x_1\nhly : ∀ i ≥ y, l ⊓ m i ≤ n i\nhlk : l ≤ m k\n⊢ n (x ⊓ y ⊔ k) = n k"
] | ← sup_inf_right, | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.Order.Lattice.Congruence | {
"line": 101,
"column": 10
} | {
"line": 101,
"column": 34
} | {
"line": 101,
"column": 35
} | [
{
"pp": "F : Type u_1\nα : Type u_2\nβ : Type u_3\ninst✝¹ : Lattice α\ninst✝ : Lattice β\nr : α → α → Prop\nh₁ : Std.Refl r\nh₂ : ∀ ⦃x y : α⦄, r x y ↔ r (x ⊓ y) (x ⊔ y)\nh₃ : ∀ ⦃x y z : α⦄, x ≤ y → y ≤ z → r x y → r y z → r x z\nh₄ : ∀ ⦃x y t : α⦄, x ≤ y → r x y → r (x ⊓ t) (y ⊓ t) ∧ r (x ⊔ t) (y ⊔ t)\nw x✝ y✝ ... | [
"F : Type u_1\nα : Type u_2\nβ : Type u_3\ninst✝¹ : Lattice α\ninst✝ : Lattice β\nr : α → α → Prop\nh₁ : Std.Refl r\nh₂ : ∀ ⦃x y : α⦄, r x y ↔ r (x ⊓ y) (x ⊔ y)\nh₃ : ∀ ⦃x y z : α⦄, x ≤ y → y ≤ z → r x y → r y z → r x z\nh₄ : ∀ ⦃x y t : α⦄, x ≤ y → r x y → r (x ⊓ t) (y ⊓ t) ∧ r (x ⊔ t) (y ⊔ t)\nw x✝ y✝ z✝ : α\nh1 :... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Order.Nucleus | {
"line": 222,
"column": 6
} | {
"line": 222,
"column": 17
} | {
"line": 222,
"column": 18
} | [
{
"pp": "X : Type u_1\ninst✝ : Frame X\nn✝ m✝ : Nucleus X\nx y : X\nm n : Nucleus X\n⊢ ∀ (x : X), x ≤ ⨅ y, ⨅ (_ : y ≥ x), m y ⇨ n y",
"ppTerm": "?m.308",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"iInf",
"CompleteLattice.toLattice",
"Iff.of_eq",
"congrArg",
"... | [
"X : Type u_1\ninst✝ : Frame X\nn✝ m✝ : Nucleus X\nx y : X\nm n : Nucleus X\n⊢ ∀ (x i : X), x ≤ i → x ⊓ m i ≤ n i"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Order.Nucleus | {
"line": 229,
"column": 4
} | {
"line": 230,
"column": 11
} | {
"line": 230,
"column": 12
} | [
{
"pp": "X : Type u_1\ninst✝ : Frame X\nn✝ m : Nucleus X\nx y : X\nx✝¹ n x✝ : Nucleus X\n⊢ x✝¹ ≤ n ⇨ x✝ ↔ x✝¹ ⊓ n ≤ x✝",
"ppTerm": "?m.18",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"Lattice.toSemilatticeSup",
"iInf",
"CompleteLattice.toLattice",
"Iff.of_eq",
... | [
"X : Type u_1\ninst✝ : Frame X\nn✝ m : Nucleus X\nx y : X\nx✝¹ n x✝ : Nucleus X\n⊢ (∀ (i i_1 : X), i ≤ i_1 → x✝¹ i ⊓ n i_1 ≤ x✝ i_1) ↔ ∀ (i : X), x✝¹ i ⊓ n i ≤ x✝ i"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Order.Lattice.Congruence | {
"line": 110,
"column": 6
} | {
"line": 110,
"column": 30
} | {
"line": 110,
"column": 31
} | [
{
"pp": "F : Type u_1\nα : Type u_2\nβ : Type u_3\ninst✝¹ : Lattice α\ninst✝ : Lattice β\nr : α → α → Prop\nh₁ : Std.Refl r\nh₂ : ∀ ⦃x y : α⦄, r x y ↔ r (x ⊓ y) (x ⊔ y)\nh₃ : ∀ ⦃x y z : α⦄, x ≤ y → y ≤ z → r x y → r y z → r x z\nh₄ : ∀ ⦃x y t : α⦄, x ≤ y → r x y → r (x ⊓ t) (y ⊓ t) ∧ r (x ⊔ t) (y ⊔ t)\nw x✝ y✝ ... | [
"F : Type u_1\nα : Type u_2\nβ : Type u_3\ninst✝¹ : Lattice α\ninst✝ : Lattice β\nr : α → α → Prop\nh₁ : Std.Refl r\nh₂ : ∀ ⦃x y : α⦄, r x y ↔ r (x ⊓ y) (x ⊔ y)\nh₃ : ∀ ⦃x y z : α⦄, x ≤ y → y ≤ z → r x y → r y z → r x z\nh₄ : ∀ ⦃x y t : α⦄, x ≤ y → r x y → r (x ⊓ t) (y ⊓ t) ∧ r (x ⊔ t) (y ⊔ t)\nw x✝ y✝ z✝ : α\nh1 :... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Order.Partition.Basic | {
"line": 88,
"column": 48
} | {
"line": 88,
"column": 59
} | {
"line": 88,
"column": 60
} | [
{
"pp": "case mk.mk\nα : Type u_1\ns✝ t x y z : α\nS : Set α\ninst✝ : CompleteLattice α\nP Q : Partition s✝\ns : α\nparts✝¹ : Set α\nsSupIndep'✝¹ : sSupIndep parts✝¹\nbot_notMem'✝¹ : ⊥ ∉ parts✝¹\nsSup_eq'✝¹ : sSup parts✝¹ = s\nparts✝ : Set α\nsSupIndep'✝ : sSupIndep parts✝\nbot_notMem'✝ : ⊥ ∉ parts✝\nsSup_eq'✝ ... | [
"case mk.mk\nα : Type u_1\ns✝ t x y z : α\nS : Set α\ninst✝ : CompleteLattice α\nP Q : Partition s✝\ns : α\nparts✝¹ : Set α\nsSupIndep'✝¹ : sSupIndep parts✝¹\nbot_notMem'✝¹ : ⊥ ∉ parts✝¹\nsSup_eq'✝¹ : sSup parts✝¹ = s\nparts✝ : Set α\nsSupIndep'✝ : sSupIndep parts✝\nbot_notMem'✝ : ⊥ ∉ parts✝\nsSup_eq'✝ : sSup parts... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Order.SaddlePoint | {
"line": 95,
"column": 2
} | {
"line": 97,
"column": 38
} | {
"line": 101,
"column": 0
} | [
{
"pp": "case refine_3\nE : Type u_1\nF : Type u_2\nβ : Type u_3\nX : Set E\nY : Set F\nf : E → F → β\ninst✝ : CompleteLinearOrder β\na : E\nha : a ∈ X\nb : F\nhb : b ∈ Y\nh : ⨆ y ∈ Y, f a y ≤ ⨅ x ∈ X, f x b\n⊢ ⨅ x ∈ X, f x b = f a b",
"ppTerm": "?refine_3",
"assigned": true,
"usedConstants": [
... | [] | · apply le_antisymm
· apply iInf₂_le a ha
· apply le_trans (le_iSup₂ b hb) h | Lean.Elab.Tactic.evalTacticCDot | Lean.cdot |
Mathlib.Order.Partition.Basic | {
"line": 410,
"column": 2
} | {
"line": 410,
"column": 37
} | {
"line": 410,
"column": 38
} | [
{
"pp": "α : Type u_1\nx : α\nu : Set α\nP : Partition u\nh : x ∈ u\n⊢ (P.partOf x).Nonempty",
"ppTerm": "?m.34",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"id",
"_private.Mathlib.Order.Partition.Basic.0.Partition.partOf_nonempty_iff._simp_1_3",
"Partition.partOf",
... | [
"α : Type u_1\nx : α\nu : Set α\nP : Partition u\nh : x ∈ u\n⊢ ¬P.partOf x = ∅"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Order.SuccPred.Tree | {
"line": 70,
"column": 4
} | {
"line": 70,
"column": 22
} | {
"line": 70,
"column": 23
} | [
{
"pp": "α : Type u_1\ninst✝⁴ : PartialOrder α\ninst✝³ : PredOrder α\ninst✝² : IsPredArchimedean α\ninst✝¹ : OrderBot α\ninst✝ : DecidableEq α\nr : α\nh : Order.pred^[Nat.find ⋯ - 1] r = ⊥\nthis : Nat.find ⋯ = 0\n⊢ r = ⊥",
"ppTerm": "?m.59",
"assigned": false,
"usedConstants": [],
"usedFVars": [... | [
"α : Type u_1\ninst✝⁴ : PartialOrder α\ninst✝³ : PredOrder α\ninst✝² : IsPredArchimedean α\ninst✝¹ : OrderBot α\ninst✝ : DecidableEq α\nr : α\nh : Order.pred^[Nat.find ⋯ - 1] r = ⊥\nthis : Nat.find ⋯ = 0\n⊢ r = ⊥"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Order.SuccPred.Tree | {
"line": 82,
"column": 4
} | {
"line": 82,
"column": 15
} | {
"line": 82,
"column": 16
} | [
{
"pp": "case right\nα : Type u_1\ninst✝⁴ : PartialOrder α\ninst✝³ : PredOrder α\ninst✝² : IsPredArchimedean α\ninst✝¹ : OrderBot α\ninst✝ : DecidableEq α\nr : α\nhr : r ≠ ⊥\nb : α\nhb : b ≤ Order.pred (findAtom r)\n⊢ b = ⊥",
"ppTerm": "?right",
"assigned": false,
"usedConstants": [],
"usedFVars... | [
"case right\nα : Type u_1\ninst✝⁴ : PartialOrder α\ninst✝³ : PredOrder α\ninst✝² : IsPredArchimedean α\ninst✝¹ : OrderBot α\ninst✝ : DecidableEq α\nr : α\nhr : r ≠ ⊥\nb : α\nhb : b ≤ Order.pred (findAtom r)\n⊢ b = ⊥"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Order.SuccPred.Tree | {
"line": 151,
"column": 4
} | {
"line": 151,
"column": 59
} | {
"line": 151,
"column": 60
} | [
{
"pp": "α : Type u_1\ninst✝² : PartialOrder α\ninst✝¹ : PredOrder α\ninst✝ : IsPredArchimedean α\nt : RootedTree\na₁ a₂ : SubRootedTree t\nh : (fun v ↦ Set.Ici v.root) a₁ = (fun v ↦ Set.Ici v.root) a₂\n⊢ a₁ = a₂",
"ppTerm": "?m.15",
"assigned": false,
"usedConstants": [],
"usedFVars": [],
"... | [
"α : Type u_1\ninst✝² : PartialOrder α\ninst✝¹ : PredOrder α\ninst✝ : IsPredArchimedean α\nt : RootedTree\na₁ a₂ : SubRootedTree t\nh : (fun v ↦ Set.Ici v.root) a₁ = (fun v ↦ Set.Ici v.root) a₂\n⊢ a₁ = a₂"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Order.SuccPred.Tree | {
"line": 205,
"column": 4
} | {
"line": 205,
"column": 81
} | {
"line": 206,
"column": 6
} | [
{
"pp": "case inr\nt : RootedTree\nt₁ t₂ : SubRootedTree t\nht₁ : t₁ ∈ t.subtrees\nht₂ : t₂ ∈ t.subtrees\nv₁ v₂ : ↑t\nh₁ : t₁.root ≤ v₁\nh₂ : t₂.root ≤ v₂\nh : ⊥ < v₁ ⊓ v₂\noh : t₁.root ≤ v₁ ∧ t₁.root ≤ v₂\n⊢ t₁.root = t₂.root",
"ppTerm": "?inr",
"assigned": false,
"usedConstants": [],
"usedFVar... | [
"case inr\nt : RootedTree\nt₁ t₂ : SubRootedTree t\nht₁ : t₁ ∈ t.subtrees\nht₂ : t₂ ∈ t.subtrees\nv₁ v₂ : ↑t\nh₁ : t₁.root ≤ v₁\nh₂ : t₂.root ≤ v₂\nh : ⊥ < v₁ ⊓ v₂\noh : t₁.root ≤ v₁ ∧ t₁.root ≤ v₂\n⊢ t₁.root = t₂.root"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Order.Sublocale | {
"line": 83,
"column": 28
} | {
"line": 83,
"column": 39
} | {
"line": 83,
"column": 40
} | [
{
"pp": "X : Type u_1\ninst✝ : Order.Frame X\nS : Sublocale X\n⊢ ⊤ ∈ S",
"ppTerm": "?m.8",
"assigned": false,
"usedConstants": [],
"usedFVars": [],
"usedGoals": []
}
] | [
"X : Type u_1\ninst✝ : Order.Frame X\nS : Sublocale X\n⊢ ⊤ ∈ S"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Order.Sublocale | {
"line": 212,
"column": 2
} | {
"line": 212,
"column": 77
} | {
"line": 213,
"column": 4
} | [
{
"pp": "X : Type u_1\ninst✝ : Order.Frame X\nn : Nucleus X\nx : X\n⊢ ↑(n.toSublocale.restrict x) = ↑⟨n x, ⋯⟩",
"ppTerm": "?m.22",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"FrameHom",
"iInf",
"congrArg",
"Nucleus",
"le_iInf_iff._simp_1",
"PartialOrder.... | [
"X : Type u_1\ninst✝ : Order.Frame X\nn : Nucleus X\nx : X\n⊢ ⨅ a, ⨅ (_ : x ≤ ↑a), ↑a ≤ n x ∧ ∀ (a : X), x ≤ n a → n x ≤ n a"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Order.Sublocale | {
"line": 213,
"column": 68
} | {
"line": 213,
"column": 79
} | {
"line": 213,
"column": 80
} | [
{
"pp": "X : Type u_1\ninst✝ : Order.Frame X\nn : Nucleus X\nx y : X\nhxy : x ≤ n y\n⊢ n x ≤ n y",
"ppTerm": "?m.54",
"assigned": false,
"usedConstants": [],
"usedFVars": [],
"usedGoals": []
}
] | [
"X : Type u_1\ninst✝ : Order.Frame X\nn : Nucleus X\nx y : X\nhxy : x ≤ n y\n⊢ n x ≤ n y"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Order.Sublocale | {
"line": 223,
"column": 27
} | {
"line": 223,
"column": 38
} | {
"line": 223,
"column": 39
} | [
{
"pp": "X : Type u_1\ninst✝ : Order.Frame X\nS : Sublocale X\nx : X\n⊢ x ∈ (fun n ↦ (OrderDual.ofDual n).toSublocale) ((fun s ↦ OrderDual.toDual s.toNucleus) S) ↔ x ∈ S",
"ppTerm": "?m.31",
"assigned": true,
"usedConstants": [
"OrderDual.toDual",
"Eq.mpr",
"FrameHom",
"Equiv... | [
"X : Type u_1\ninst✝ : Order.Frame X\nS : Sublocale X\nx : X\n⊢ (∃ y, ↑(S.restrict y) = x) ↔ x ∈ S"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Order.Types.Arithmetic | {
"line": 122,
"column": 45
} | {
"line": 122,
"column": 56
} | {
"line": 122,
"column": 57
} | [
{
"pp": "⊢ card 0 = 0",
"ppTerm": "?m.6",
"assigned": false,
"usedConstants": [],
"usedFVars": [],
"usedGoals": []
}
] | [
"⊢ card 0 = 0"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Order.Types.Arithmetic | {
"line": 124,
"column": 44
} | {
"line": 124,
"column": 55
} | {
"line": 124,
"column": 56
} | [
{
"pp": "⊢ card 1 = 1",
"ppTerm": "?m.6",
"assigned": false,
"usedConstants": [],
"usedFVars": [],
"usedGoals": []
}
] | [
"⊢ card 1 = 1"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Probability.Martingale.Centering | {
"line": 100,
"column": 2
} | {
"line": 100,
"column": 36
} | {
"line": 100,
"column": 37
} | [
{
"pp": "Ω : Type u_1\nE : Type u_2\nm0 : MeasurableSpace Ω\nμ : Measure Ω\ninst✝⁴ : NormedAddCommGroup E\ninst✝³ : NormedSpace ℝ E\nf : ℕ → Ω → E\nℱ : Filtration ℕ m0\ninst✝² : CompleteSpace E\ninst✝¹ : PartialOrder E\ninst✝ : IsOrderedAddMonoid E\nhf : Submartingale f ℱ μ\nω : Ω\nhω : Monotone fun x ↦ predict... | [
"Ω : Type u_1\nE : Type u_2\nm0 : MeasurableSpace Ω\nμ : Measure Ω\ninst✝⁴ : NormedAddCommGroup E\ninst✝³ : NormedSpace ℝ E\nf : ℕ → Ω → E\nℱ : Filtration ℕ m0\ninst✝² : CompleteSpace E\ninst✝¹ : PartialOrder E\ninst✝ : IsOrderedAddMonoid E\nhf : Submartingale f ℱ μ\nω : Ω\nhω : Monotone fun x ↦ predictablePart f ℱ... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Probability.Martingale.Centering | {
"line": 257,
"column": 2
} | {
"line": 257,
"column": 57
} | {
"line": 258,
"column": 4
} | [
{
"pp": "Ω : Type u_1\nE : Type u_2\nm0 : MeasurableSpace Ω\nμ : Measure Ω\ninst✝² : NormedAddCommGroup E\ninst✝¹ : NormedSpace ℝ E\ninst✝ : CompleteSpace E\nR : ℝ\nf : ℕ → Ω → E\nℱ : Filtration ℕ m0\nhbdd : ∀ᵐ (ω : Ω) ∂μ, ∀ (i : ℕ), ‖f (i + 1) ω - f i ω‖ ≤ R\nω : Ω\nhω₁ : ∀ (i : ℕ), ‖f (i + 1) ω - f i ω‖ ≤ R\n... | [
"Ω : Type u_1\nE : Type u_2\nm0 : MeasurableSpace Ω\nμ : Measure Ω\ninst✝² : NormedAddCommGroup E\ninst✝¹ : NormedSpace ℝ E\ninst✝ : CompleteSpace E\nR : ℝ\nf : ℕ → Ω → E\nℱ : Filtration ℕ m0\nhbdd : ∀ᵐ (ω : Ω) ∂μ, ∀ (i : ℕ), ‖f (i + 1) ω - f i ω‖ ≤ R\nω : Ω\nhω₁ : ∀ (i : ℕ), ‖f (i + 1) ω - f i ω‖ ≤ R\nhω₂ : ∀ (i :... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Probability.Martingale.Basic | {
"line": 347,
"column": 2
} | {
"line": 347,
"column": 38
} | {
"line": 347,
"column": 39
} | [
{
"pp": "Ω : Type u_1\nι : Type u_3\ninst✝⁵ : Preorder ι\nm0 : MeasurableSpace Ω\nμ : Measure Ω\nℱ : Filtration ι m0\nF : Type u_4\ninst✝⁴ : NormedAddCommGroup F\ninst✝³ : PartialOrder F\ninst✝² : NormedSpace ℝ F\ninst✝¹ : CompleteSpace F\ninst✝ : IsOrderedModule ℝ F\nf : ι → Ω → F\nc : ℝ\nhc : 0 ≤ c\nhf : Supe... | [
"Ω : Type u_1\nι : Type u_3\ninst✝⁵ : Preorder ι\nm0 : MeasurableSpace Ω\nμ : Measure Ω\nℱ : Filtration ι m0\nF : Type u_4\ninst✝⁴ : NormedAddCommGroup F\ninst✝³ : PartialOrder F\ninst✝² : NormedSpace ℝ F\ninst✝¹ : CompleteSpace F\ninst✝ : IsOrderedModule ℝ F\nf : ι → Ω → F\nc : ℝ\nhc : 0 ≤ c\nhf : Supermartingale ... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Probability.Process.HittingTime | {
"line": 163,
"column": 2
} | {
"line": 163,
"column": 13
} | {
"line": 165,
"column": 0
} | [
{
"pp": "case neg\nΩ : Type u_1\nβ : Type u_2\nι : Type u_3\ninst✝ : ConditionallyCompleteLinearOrder ι\nu : ι → Ω → β\ns : Set β\nn m : ι\nhnm : n ≤ m\nω : Ω\nh : ¬∃ j ∈ Set.Icc n m, u j ω ∈ s\n⊢ n ≤ m",
"ppTerm": "?neg✝",
"assigned": true,
"usedConstants": [],
"usedFVars": [
"hnm"
],... | [] | · exact hnm | Lean.Elab.Tactic.evalTacticCDot | Lean.cdot |
Mathlib.Probability.Martingale.Basic | {
"line": 393,
"column": 2
} | {
"line": 395,
"column": 62
} | {
"line": 397,
"column": 0
} | [
{
"pp": "Ω : Type u_1\nm0 : MeasurableSpace Ω\nμ : Measure Ω\n𝒢 : Filtration ℕ m0\ninst✝ : IsFiniteMeasure μ\nf : ℕ → Ω → ℝ\nhadp : StronglyAdapted 𝒢 f\nhint : ∀ (i : ℕ), Integrable (f i) μ\nhf : ∀ (i : ℕ) (s : Set Ω), MeasurableSet s → ∫ (ω : Ω) in s, f i ω ∂μ ≤ ∫ (ω : Ω) in s, f (i + 1) ω ∂μ\ni j : ℕ\nhij :... | [] | induction hij with
| refl => rfl
| step hk₁ hk₂ => exact hk₂.trans (hf _ s (𝒢.mono hk₁ _ hs)) | _private.Lean.Elab.Tactic.Induction.0.Lean.Elab.Tactic.evalInduction | Lean.Parser.Tactic.induction |
Mathlib.Probability.Martingale.Convergence | {
"line": 150,
"column": 2
} | {
"line": 152,
"column": 99
} | {
"line": 154,
"column": 0
} | [
{
"pp": "case neg\nΩ : Type u_1\nf : ℕ → Ω → ℝ\nω : Ω\nhf₁ : liminf (fun n ↦ ↑‖f n ω‖₊) atTop < ∞\nhf₂ : ∀ (a b : ℚ), a < b → upcrossings (↑a) (↑b) f ω < ∞\nh : ¬IsBoundedUnder (fun x1 x2 ↦ x1 ≤ x2) atTop fun n ↦ |f n ω|\n⊢ ∃ c, Tendsto (fun n ↦ f n ω) atTop (𝓝 c)",
"ppTerm": "?neg✝",
"assigned": true,... | [] | · obtain ⟨a, b, hab, h₁, h₂⟩ := ENNReal.exists_upcrossings_of_not_bounded_under hf₁.ne h
exact
False.elim ((hf₂ a b hab).ne (upcrossings_eq_top_of_frequently_lt (Rat.cast_lt.2 hab) h₁ h₂)) | Lean.Elab.Tactic.evalTacticCDot | Lean.cdot |
Mathlib.Probability.Martingale.Upcrossing | {
"line": 495,
"column": 2
} | {
"line": 503,
"column": 16
} | {
"line": 505,
"column": 0
} | [
{
"pp": "Ω : Type u_1\na b : ℝ\nf : ℕ → Ω → ℝ\nN n : ℕ\nω : Ω\nM : ℕ\nhNM : N ≤ M\nh : upperCrossingTime a b f N (n + 1) ω < N\n⊢ upperCrossingTime a b f M (n + 1) ω = upperCrossingTime a b f N (n + 1) ω ∧\n lowerCrossingTime a b f M n ω = lowerCrossingTime a b f N n ω",
"ppTerm": "?m.53",
"assigned"... | [] | have := (crossing_eq_crossing_of_lowerCrossingTime_lt hNM
(lt_of_le_of_lt lowerCrossingTime_le_upperCrossingTime_succ h)).2
refine ⟨?_, this⟩
rw [upperCrossingTime_succ_eq, upperCrossingTime_succ_eq, eq_comm, this]
refine hittingBtwn_eq_hittingBtwn_of_exists hNM ?_
rw [upperCrossingTime_succ_eq, hittingBtwn... | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.Probability.Martingale.Upcrossing | {
"line": 495,
"column": 2
} | {
"line": 503,
"column": 16
} | {
"line": 505,
"column": 0
} | [
{
"pp": "Ω : Type u_1\na b : ℝ\nf : ℕ → Ω → ℝ\nN n : ℕ\nω : Ω\nM : ℕ\nhNM : N ≤ M\nh : upperCrossingTime a b f N (n + 1) ω < N\n⊢ upperCrossingTime a b f M (n + 1) ω = upperCrossingTime a b f N (n + 1) ω ∧\n lowerCrossingTime a b f M n ω = lowerCrossingTime a b f N n ω",
"ppTerm": "?m.53",
"assigned"... | [] | have := (crossing_eq_crossing_of_lowerCrossingTime_lt hNM
(lt_of_le_of_lt lowerCrossingTime_le_upperCrossingTime_succ h)).2
refine ⟨?_, this⟩
rw [upperCrossingTime_succ_eq, upperCrossingTime_succ_eq, eq_comm, this]
refine hittingBtwn_eq_hittingBtwn_of_exists hNM ?_
rw [upperCrossingTime_succ_eq, hittingBtwn... | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.Probability.Martingale.Convergence | {
"line": 335,
"column": 2
} | {
"line": 338,
"column": 50
} | {
"line": 339,
"column": 2
} | [
{
"pp": "Ω : Type u_1\nm0 : MeasurableSpace Ω\nℱ : Filtration ℕ m0\nf : ℕ → Ω → ℝ\ng : Ω → ℝ\nμ : Measure Ω\nhf : Martingale f ℱ μ\nhg : Integrable g μ\nhgtends : Tendsto (fun n ↦ eLpNorm (f n - g) 1 μ) atTop (𝓝 0)\nn : ℕ\n⊢ eLpNorm (f n - μ[g | ↑ℱ n]) 1 μ = 0",
"ppTerm": "?m.126",
"assigned": true,
... | [
"Ω : Type u_1\nm0 : MeasurableSpace Ω\nℱ : Filtration ℕ m0\nf : ℕ → Ω → ℝ\ng : Ω → ℝ\nμ : Measure Ω\nhf : Martingale f ℱ μ\nhg : Integrable g μ\nhgtends : Tendsto (fun n ↦ eLpNorm (f n - g) 1 μ) atTop (𝓝 0)\nn : ℕ\nht : Tendsto (fun m ↦ eLpNorm μ[f m - g | ↑ℱ n] 1 μ) atTop (𝓝 0)\n⊢ eLpNorm (f n - μ[g | ↑ℱ n]) 1 μ... | have ht : Tendsto (fun m => eLpNorm (μ[f m - g | ℱ n]) 1 μ) atTop (𝓝 0) :=
haveI hint : ∀ m, Integrable (f m - g) μ := fun m => (hf.integrable m).sub hg
tendsto_of_tendsto_of_tendsto_of_le_of_le tendsto_const_nhds hgtends (fun m => zero_le)
fun m => eLpNorm_condExp_le_eLpNorm _ le_rfl | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_tacticHave___1 | Lean.Parser.Tactic.tacticHave__ |
Mathlib.Probability.Process.HittingTime | {
"line": 411,
"column": 4
} | {
"line": 411,
"column": 32
} | {
"line": 411,
"column": 33
} | [
{
"pp": "case inr\nΩ : Type u_1\nβ : Type u_2\nι : Type u_3\nm : MeasurableSpace Ω\ninst✝² : ConditionallyCompleteLinearOrder ι\ninst✝¹ : WellFoundedLT ι\ninst✝ : Countable ι\nx✝ : MeasurableSpace β\nf : Filtration ι m\nu : ι → Ω → β\ns : Set β\nn n' : ι\nhu : Adapted f u\nhs : MeasurableSet s\ni : ι\nhi : i < ... | [
"case inr\nΩ : Type u_1\nβ : Type u_2\nι : Type u_3\nm : MeasurableSpace Ω\ninst✝² : ConditionallyCompleteLinearOrder ι\ninst✝¹ : WellFoundedLT ι\ninst✝ : Countable ι\nx✝ : MeasurableSpace β\nf : Filtration ι m\nu : ι → Ω → β\ns : Set β\nn n' : ι\nhu : Adapted f u\nhs : MeasurableSet s\ni : ι\nhi : i < n'\nh_set_eq... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Probability.Process.HittingTime | {
"line": 424,
"column": 2
} | {
"line": 424,
"column": 30
} | {
"line": 424,
"column": 31
} | [
{
"pp": "Ω : Type u_1\nβ : Type u_2\nι : Type u_3\nm : MeasurableSpace Ω\ninst✝² : ConditionallyCompleteLinearOrder ι\ninst✝¹ : WellFoundedLT ι\ninst✝ : Countable ι\nx✝ : MeasurableSpace β\nf : Filtration ι m\nu : ι → Ω → β\ns : Set β\nn : ι\nhu : Adapted f u\nhs : MeasurableSet s\ni : ι\nh_set_eq_Union : {ω | ... | [
"Ω : Type u_1\nβ : Type u_2\nι : Type u_3\nm : MeasurableSpace Ω\ninst✝² : ConditionallyCompleteLinearOrder ι\ninst✝¹ : WellFoundedLT ι\ninst✝ : Countable ι\nx✝ : MeasurableSpace β\nf : Filtration ι m\nu : ι → Ω → β\ns : Set β\nn : ι\nhu : Adapted f u\nhs : MeasurableSet s\ni : ι\nh_set_eq_Union : {ω | hittingAfter... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Probability.Process.HittingTime | {
"line": 421,
"column": 2
} | {
"line": 425,
"column": 60
} | {
"line": 427,
"column": 0
} | [
{
"pp": "Ω : Type u_1\nβ : Type u_2\nι : Type u_3\nm : MeasurableSpace Ω\ninst✝² : ConditionallyCompleteLinearOrder ι\ninst✝¹ : WellFoundedLT ι\ninst✝ : Countable ι\nx✝ : MeasurableSpace β\nf : Filtration ι m\nu : ι → Ω → β\ns : Set β\nn : ι\nhu : Adapted f u\nhs : MeasurableSet s\n⊢ IsStoppingTime f (hittingAf... | [] | intro i
have h_set_eq_Union : {ω | hittingAfter u s n ω ≤ i} = ⋃ j ∈ Set.Icc n i, u j ⁻¹' s := by
ext; simp [hittingAfter_le_iff]
simpa [h_set_eq_Union] using MeasurableSet.iUnion fun j =>
MeasurableSet.iUnion fun hj => f.mono hj.2 _ ((hu j) hs) | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.Probability.Process.HittingTime | {
"line": 421,
"column": 2
} | {
"line": 425,
"column": 60
} | {
"line": 427,
"column": 0
} | [
{
"pp": "Ω : Type u_1\nβ : Type u_2\nι : Type u_3\nm : MeasurableSpace Ω\ninst✝² : ConditionallyCompleteLinearOrder ι\ninst✝¹ : WellFoundedLT ι\ninst✝ : Countable ι\nx✝ : MeasurableSpace β\nf : Filtration ι m\nu : ι → Ω → β\ns : Set β\nn : ι\nhu : Adapted f u\nhs : MeasurableSet s\n⊢ IsStoppingTime f (hittingAf... | [] | intro i
have h_set_eq_Union : {ω | hittingAfter u s n ω ≤ i} = ⋃ j ∈ Set.Icc n i, u j ⁻¹' s := by
ext; simp [hittingAfter_le_iff]
simpa [h_set_eq_Union] using MeasurableSet.iUnion fun j =>
MeasurableSet.iUnion fun hj => f.mono hj.2 _ ((hu j) hs) | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.Probability.Process.Stopping | {
"line": 435,
"column": 28
} | {
"line": 435,
"column": 39
} | {
"line": 435,
"column": 40
} | [
{
"pp": "case coe.coe\nΩ : Type u_1\nι : Type u_3\nm : MeasurableSpace Ω\ninst✝⁵ : Add ι\ninst✝⁴ : LinearOrder ι\ninst✝³ : CanonicallyOrderedAdd ι\ninst✝² : Countable ι\ninst✝¹ : TopologicalSpace ι\ninst✝ : OrderTopology ι\nf : Filtration ι m\nτ π : Ω → WithTop ι\nhτ : IsStoppingTime f τ\nhπ : IsStoppingTime f ... | [
"case coe.coe\nΩ : Type u_1\nι : Type u_3\nm : MeasurableSpace Ω\ninst✝⁵ : Add ι\ninst✝⁴ : LinearOrder ι\ninst✝³ : CanonicallyOrderedAdd ι\ninst✝² : Countable ι\ninst✝¹ : TopologicalSpace ι\ninst✝ : OrderTopology ι\nf : Filtration ι m\nτ π : Ω → WithTop ι\nhτ : IsStoppingTime f τ\nhπ : IsStoppingTime f π\nj : ι\nω ... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Probability.Process.HittingTime | {
"line": 471,
"column": 2
} | {
"line": 471,
"column": 13
} | {
"line": 471,
"column": 14
} | [
{
"pp": "Ω : Type u_1\nβ : Type u_2\nι : Type u_3\nm : MeasurableSpace Ω\ninst✝⁶ : ConditionallyCompleteLinearOrder ι\ninst✝⁵ : WellFoundedLT ι\ninst✝⁴ : Countable ι\ninst✝³ : TopologicalSpace ι\ninst✝² : OrderTopology ι\ninst✝¹ : FirstCountableTopology ι\ninst✝ : MeasurableSpace β\nf : Filtration ι m\nu : ι → ... | [
"Ω : Type u_1\nβ : Type u_2\nι : Type u_3\nm : MeasurableSpace Ω\ninst✝⁶ : ConditionallyCompleteLinearOrder ι\ninst✝⁵ : WellFoundedLT ι\ninst✝⁴ : Countable ι\ninst✝³ : TopologicalSpace ι\ninst✝² : OrderTopology ι\ninst✝¹ : FirstCountableTopology ι\ninst✝ : MeasurableSpace β\nf : Filtration ι m\nu : ι → Ω → β\nτ : Ω... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Probability.Process.Stopping | {
"line": 487,
"column": 4
} | {
"line": 487,
"column": 62
} | {
"line": 487,
"column": 63
} | [
{
"pp": "case mp\nΩ : Type u_1\nι : Type u_3\nm : MeasurableSpace Ω\ninst✝ : Preorder ι\nf : Filtration ι m\ni : ι\ns : Set Ω\nh : MeasurableSet s ∧ ∀ (i_1 : ι), MeasurableSet (s ∩ {ω | ↑i ≤ ↑i_1})\nh' : MeasurableSet (s ∩ {ω | ↑i ≤ ↑i})\n⊢ MeasurableSet s",
"ppTerm": "?mp",
"assigned": false,
"used... | [
"case mp\nΩ : Type u_1\nι : Type u_3\nm : MeasurableSpace Ω\ninst✝ : Preorder ι\nf : Filtration ι m\ni : ι\ns : Set Ω\nh : MeasurableSet s ∧ ∀ (i_1 : ι), MeasurableSet (s ∩ {ω | ↑i ≤ ↑i_1})\nh' : MeasurableSet (s ∩ {ω | ↑i ≤ ↑i})\n⊢ MeasurableSet s"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Probability.Process.Stopping | {
"line": 506,
"column": 4
} | {
"line": 506,
"column": 39
} | {
"line": 506,
"column": 40
} | [
{
"pp": "case mp\nΩ : Type u_1\nι : Type u_3\nm : MeasurableSpace Ω\ninst✝ : Preorder ι\nf : Filtration ι m\nτ : Ω → WithTop ι\nhτ : IsStoppingTime f τ\ns : Set Ω\ni : ι\nthis : ∀ (j : WithTop ι), {ω | τ ω = ↑i} ∩ {ω | τ ω ≤ j} = {ω | τ ω = ↑i} ∩ {_ω | ↑i ≤ j}\nh : MeasurableSet (s ∩ {ω | τ ω = ↑i})\n⊢ Measurab... | [
"case mp\nΩ : Type u_1\nι : Type u_3\nm : MeasurableSpace Ω\ninst✝ : Preorder ι\nf : Filtration ι m\nτ : Ω → WithTop ι\nhτ : IsStoppingTime f τ\ns : Set Ω\ni : ι\nthis : ∀ (j : WithTop ι), {ω | τ ω = ↑i} ∩ {ω | τ ω ≤ j} = {ω | τ ω = ↑i} ∩ {_ω | ↑i ≤ j}\nh : MeasurableSet (s ∩ {ω | τ ω = ↑i})\n⊢ MeasurableSet (s ∩ {... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Probability.Martingale.Upcrossing | {
"line": 591,
"column": 6
} | {
"line": 613,
"column": 47
} | {
"line": 614,
"column": 2
} | [] | [] | ∑ _k ∈ Finset.range (upcrossingsBefore a b f N ω), (b - a) ≤
∑ k ∈ Finset.range (upcrossingsBefore a b f N ω),
(stoppedValue f (fun ω ↦ (upperCrossingTime a b f N (k + 1) ω : ℕ)) ω -
stoppedValue f (fun ω ↦ (lowerCrossingTime a b f N k ω : ℕ)) ω) := by
gcongr ∑ k ∈ _, ?_ with... | Lean.Elab.Tactic._aux_Mathlib_Tactic_Widget_Calc___elabRules_Lean_calcTactic_1 | Lean.calcSteps |
Mathlib.Probability.Martingale.Upcrossing | {
"line": 627,
"column": 8
} | {
"line": 627,
"column": 19
} | {
"line": 627,
"column": 20
} | [
{
"pp": "case refine_2\nΩ : Type u_1\nm0 : MeasurableSpace Ω\nμ : Measure Ω\na b : ℝ\nf : ℕ → Ω → ℝ\nN : ℕ\nℱ : Filtration ℕ m0\ninst✝ : IsFiniteMeasure μ\nhf : Submartingale f ℱ μ\nhfN : ∀ (ω : Ω), a ≤ f N ω\nhfzero : 0 ≤ f 0\nhab : a < b\nω : Ω\n⊢ (b - a) * ↑(upcrossingsBefore a b f N ω) ≤ (∑ k ∈ Finset.range... | [
"case refine_2\nΩ : Type u_1\nm0 : MeasurableSpace Ω\nμ : Measure Ω\na b : ℝ\nf : ℕ → Ω → ℝ\nN : ℕ\nℱ : Filtration ℕ m0\ninst✝ : IsFiniteMeasure μ\nhf : Submartingale f ℱ μ\nhfN : ∀ (ω : Ω), a ≤ f N ω\nhfzero : 0 ≤ f 0\nhab : a < b\nω : Ω\n⊢ (b - a) * ↑(upcrossingsBefore a b f N ω) ≤ ∑ x ∈ Finset.range N, upcrossin... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Probability.Process.Stopping | {
"line": 736,
"column": 4
} | {
"line": 736,
"column": 15
} | {
"line": 736,
"column": 16
} | [
{
"pp": "Ω : Type u_1\nι : Type u_3\nm : MeasurableSpace Ω\ninst✝³ : LinearOrder ι\nf : Filtration ι m\nτ π : Ω → WithTop ι\ninst✝² : TopologicalSpace ι\ninst✝¹ : SecondCountableTopology ι\ninst✝ : OrderTopology ι\nhτ : IsStoppingTime f τ\nhπ : IsStoppingTime f π\nj : ι\nx✝ : Ω\n⊢ x✝ ∈ {ω | τ ω ≤ π ω} ∩ {ω | τ ... | [
"Ω : Type u_1\nι : Type u_3\nm : MeasurableSpace Ω\ninst✝³ : LinearOrder ι\nf : Filtration ι m\nτ π : Ω → WithTop ι\ninst✝² : TopologicalSpace ι\ninst✝¹ : SecondCountableTopology ι\ninst✝ : OrderTopology ι\nhτ : IsStoppingTime f τ\nhπ : IsStoppingTime f π\nj : ι\nx✝ : Ω\n⊢ τ x✝ ≤ ↑j → ↑j ≤ π x✝ → τ x✝ ≤ π x✝"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Probability.Martingale.OptionalStopping | {
"line": 192,
"column": 8
} | {
"line": 192,
"column": 41
} | {
"line": 193,
"column": 8
} | [
{
"pp": "case hg\nΩ : Type u_1\nm0 : MeasurableSpace Ω\nμ : Measure Ω\n𝒢 : Filtration ℕ m0\nf : ℕ → Ω → ℝ\ninst✝ : IsFiniteMeasure μ\nhsub : Submartingale f 𝒢 μ\nhnonneg : 0 ≤ f\nε : ℝ≥0\nn : ℕ\n⊢ IntegrableOn (fun ω ↦ stoppedValue f (fun ω ↦ ↑(hittingBtwn f {y | ↑ε ≤ y} 0 n ω)) ω)\n {ω | ((range (n + 1)).... | [
"case hg\nΩ : Type u_1\nm0 : MeasurableSpace Ω\nμ : Measure Ω\n𝒢 : Filtration ℕ m0\nf : ℕ → Ω → ℝ\ninst✝ : IsFiniteMeasure μ\nhsub : Submartingale f 𝒢 μ\nhnonneg : 0 ≤ f\nε : ℝ≥0\nn : ℕ\n⊢ Integrable (fun ω ↦ stoppedValue f (fun ω ↦ ↑(hittingBtwn f {y | ↑ε ≤ y} 0 n ω)) ω) μ"
] | refine Integrable.integrableOn ?_ | Lean.Elab.Tactic.evalRefine | Lean.Parser.Tactic.refine |
Mathlib.Probability.Martingale.BorelCantelli | {
"line": 207,
"column": 4
} | {
"line": 209,
"column": 39
} | {
"line": 210,
"column": 2
} | [
{
"pp": "case mpr\nΩ : Type u_2\nm0 : MeasurableSpace Ω\nμ : Measure Ω\nℱ : Filtration ℕ m0\nf : ℕ → Ω → ℝ\nR : ℝ≥0\ninst✝ : IsFiniteMeasure μ\nhf : Martingale f ℱ μ\nhbdd : ∀ᵐ (ω : Ω) ∂μ, ∀ (i : ℕ), |f (i + 1) ω - f i ω| ≤ ↑R\nhbdd' : ∀ᵐ (ω : Ω) ∂μ, ∀ (i : ℕ), |(-f) (i + 1) ω - (-f) i ω| ≤ ↑R\nhup : ∀ᵐ (ω : Ω)... | [] | · refine ⟨-c, ?_⟩
convert! hc.neg
simp only [neg_neg, Pi.neg_apply] | Lean.Elab.Tactic.evalTacticCDot | Lean.cdot |
Mathlib.Probability.ConditionalExpectation | {
"line": 58,
"column": 6
} | {
"line": 58,
"column": 21
} | {
"line": 58,
"column": 22
} | [
{
"pp": "case pos.refine_2\nΩ : Type u_1\nE : Type u_2\ninst✝³ : NormedAddCommGroup E\ninst✝² : NormedSpace ℝ E\ninst✝¹ : CompleteSpace E\nm₁ m₂ m : MeasurableSpace Ω\nμ : Measure Ω\nf : Ω → E\nhle₁ : m₁ ≤ m\nhle₂ : m₂ ≤ m\ninst✝ : SigmaFinite (μ.trim hle₂)\nhf : StronglyMeasurable f\nhindp : ∀ (t1 t2 : Set Ω),... | [
"case pos.refine_2\nΩ : Type u_1\nE : Type u_2\ninst✝³ : NormedAddCommGroup E\ninst✝² : NormedSpace ℝ E\ninst✝¹ : CompleteSpace E\nm₁ m₂ m : MeasurableSpace Ω\nμ : Measure Ω\nf : Ω → E\nhle₁ : m₁ ≤ m\nhle₂ : m₂ ≤ m\ninst✝ : SigmaFinite (μ.trim hle₂)\nhf : StronglyMeasurable f\nhindp : ∀ (t1 t2 : Set Ω), MeasurableS... | Set.inter_comm, | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.Probability.Martingale.Upcrossing | {
"line": 826,
"column": 4
} | {
"line": 828,
"column": 17
} | {
"line": 830,
"column": 0
} | [
{
"pp": "case neg\nΩ : Type u_1\nm0 : MeasurableSpace Ω\nμ : Measure Ω\nf : ℕ → Ω → ℝ\nℱ : Filtration ℕ m0\ninst✝ : IsFiniteMeasure μ\na b : ℝ\nhf : Submartingale f ℱ μ\nhab : b ≤ a\n⊢ ENNReal.ofReal (b - a) * ∫⁻ (ω : Ω), upcrossings a b f ω ∂μ ≤ ⨆ N, ∫⁻ (ω : Ω), ENNReal.ofReal (f N ω - a)⁺ ∂μ",
"ppTerm": "... | [] | rw [← sub_nonpos] at hab
rw [ENNReal.ofReal_of_nonpos hab, zero_mul]
exact zero_le | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.Probability.Martingale.Upcrossing | {
"line": 826,
"column": 4
} | {
"line": 828,
"column": 17
} | {
"line": 830,
"column": 0
} | [
{
"pp": "case neg\nΩ : Type u_1\nm0 : MeasurableSpace Ω\nμ : Measure Ω\nf : ℕ → Ω → ℝ\nℱ : Filtration ℕ m0\ninst✝ : IsFiniteMeasure μ\na b : ℝ\nhf : Submartingale f ℱ μ\nhab : b ≤ a\n⊢ ENNReal.ofReal (b - a) * ∫⁻ (ω : Ω), upcrossings a b f ω ∂μ ≤ ⨆ N, ∫⁻ (ω : Ω), ENNReal.ofReal (f N ω - a)⁺ ∂μ",
"ppTerm": "... | [] | rw [← sub_nonpos] at hab
rw [ENNReal.ofReal_of_nonpos hab, zero_mul]
exact zero_le | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.Probability.Martingale.BorelCantelli | {
"line": 326,
"column": 2
} | {
"line": 326,
"column": 13
} | {
"line": 326,
"column": 14
} | [
{
"pp": "case refine_2\nΩ : Type u_2\nm0 : MeasurableSpace Ω\nμ : Measure Ω\nℱ : Filtration ℕ m0\ninst✝ : IsFiniteMeasure μ\ns : ℕ → Set Ω\nhs : ∀ (n : ℕ), MeasurableSet (s n)\nthis :\n ∀ᵐ (ω : Ω) ∂μ,\n Tendsto (fun n ↦ (∑ k ∈ Finset.range n, (s (k + 1)).indicator 1) ω) atTop atTop ↔\n Tendsto (fun n ↦... | [
"case refine_2\nΩ : Type u_2\nm0 : MeasurableSpace Ω\nμ : Measure Ω\nℱ : Filtration ℕ m0\ninst✝ : IsFiniteMeasure μ\ns : ℕ → Set Ω\nhs : ∀ (n : ℕ), MeasurableSet (s n)\nthis :\n ∀ᵐ (ω : Ω) ∂μ,\n Tendsto (fun n ↦ (∑ k ∈ Finset.range n, (s (k + 1)).indicator 1) ω) atTop atTop ↔\n Tendsto (fun n ↦ (∑ k ∈ Fins... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Probability.Moments.ComplexMGF | {
"line": 78,
"column": 2
} | {
"line": 78,
"column": 32
} | {
"line": 78,
"column": 33
} | [
{
"pp": "Ω : Type u_1\nm : MeasurableSpace Ω\nX : Ω → ℝ\nμ : Measure Ω\nz : ℂ\nhX : AEMeasurable X μ\nh : ¬Integrable (fun ω ↦ rexp (z.re * X ω)) μ\n⊢ ¬Integrable (fun a ↦ ‖cexp (z * ↑(X a))‖) μ",
"ppTerm": "?m.135",
"assigned": true,
"usedConstants": [
"Norm.norm",
"Eq.mpr",
"Norm... | [
"Ω : Type u_1\nm : MeasurableSpace Ω\nX : Ω → ℝ\nμ : Measure Ω\nz : ℂ\nhX : AEMeasurable X μ\nh : ¬Integrable (fun ω ↦ rexp (z.re * X ω)) μ\n⊢ ¬Integrable (fun a ↦ rexp (z.re * X a)) μ"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Probability.Moments.IntegrableExpMul | {
"line": 126,
"column": 2
} | {
"line": 129,
"column": 36
} | {
"line": 130,
"column": 2
} | [
{
"pp": "case refine_1\nΩ✝ : Type u_1\nm✝ : MeasurableSpace Ω✝\nX✝ : Ω✝ → ℝ\nμ✝ : Measure Ω✝\nΩ : Type u_1\nm : MeasurableSpace Ω\nX : Ω → ℝ\nμ : Measure Ω\nt₁ : ℝ\nht₁ : t₁ ∈ integrableExpSet X μ\nt₂ : ℝ\nht₂ : t₂ ∈ integrableExpSet X μ\na b : ℝ\nha : 0 ≤ a\nhb : 0 ≤ b\nhab : a + b = 1\nh_le : t₁ ≤ t₂\n⊢ t₁ ≤ ... | [
"case refine_2\nΩ✝ : Type u_1\nm✝ : MeasurableSpace Ω✝\nX✝ : Ω✝ → ℝ\nμ✝ : Measure Ω✝\nΩ : Type u_1\nm : MeasurableSpace Ω\nX : Ω → ℝ\nμ : Measure Ω\nt₁ : ℝ\nht₁ : t₁ ∈ integrableExpSet X μ\nt₂ : ℝ\nht₂ : t₂ ∈ integrableExpSet X μ\na b : ℝ\nha : 0 ≤ a\nhb : 0 ≤ b\nhab : a + b = 1\nh_le : t₁ ≤ t₂\n⊢ a • t₁ + b • t₂ ≤... | · simp only [smul_eq_mul]
calc t₁
_ = a * t₁ + b * t₁ := by rw [← add_mul, hab, one_mul]
_ ≤ a * t₁ + b * t₂ := by gcongr | Lean.Elab.Tactic.evalTacticCDot | Lean.cdot |
Mathlib.Probability.Moments.ComplexMGF | {
"line": 130,
"column": 2
} | {
"line": 130,
"column": 40
} | {
"line": 131,
"column": 2
} | [
{
"pp": "Ω : Type u_1\nm : MeasurableSpace Ω\nX : Ω → ℝ\nμ : Measure Ω\nz : ℂ\nhz : z.re ∈ interior (integrableExpSet X μ)\nn : ℕ\nhX : AEMeasurable X μ\nl u : ℝ\nhlu : z.re ∈ Set.Ioo l u\nh_subset : Set.Ioo l u ⊆ integrableExpSet X μ\n⊢ HasDerivAt (fun z ↦ ∫ (x : Ω), (fun ω ↦ ↑(X ω) ^ n * cexp (z * ↑(X ω))) x ... | [
"Ω : Type u_1\nm : MeasurableSpace Ω\nX : Ω → ℝ\nμ : Measure Ω\nz : ℂ\nhz : z.re ∈ interior (integrableExpSet X μ)\nn : ℕ\nhX : AEMeasurable X μ\nl u : ℝ\nhlu : z.re ∈ Set.Ioo l u\nh_subset : Set.Ioo l u ⊆ integrableExpSet X μ\nt : ℝ := min (z.re - l) (u - z.re) / 2\n⊢ HasDerivAt (fun z ↦ ∫ (x : Ω), (fun ω ↦ ↑(X ω)... | let t := ((z.re - l) ⊓ (u - z.re)) / 2 | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_tacticLet___1 | Lean.Parser.Tactic.tacticLet__ |
Mathlib.Probability.Moments.IntegrableExpMul | {
"line": 154,
"column": 25
} | {
"line": 154,
"column": 36
} | {
"line": 154,
"column": 37
} | [
{
"pp": "Ω : Type u_1\nm : MeasurableSpace Ω\nX : Ω → ℝ\nμ : Measure Ω\nt v : ℝ\nht_int_pos : Integrable (fun ω ↦ rexp ((v + t) * X ω)) μ\nht_int_neg : Integrable (fun ω ↦ rexp ((v - t) * X ω)) μ\n⊢ Integrable (fun a ↦ rexp ((v - t) * X a)) μ",
"ppTerm": "?m.87",
"assigned": false,
"usedConstants": ... | [
"Ω : Type u_1\nm : MeasurableSpace Ω\nX : Ω → ℝ\nμ : Measure Ω\nt v : ℝ\nht_int_pos : Integrable (fun ω ↦ rexp ((v + t) * X ω)) μ\nht_int_neg : Integrable (fun ω ↦ rexp ((v - t) * X ω)) μ\n⊢ Integrable (fun a ↦ rexp ((v - t) * X a)) μ"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Probability.Process.Stopping | {
"line": 1097,
"column": 4
} | {
"line": 1097,
"column": 15
} | {
"line": 1097,
"column": 16
} | [
{
"pp": "case mpr\nΩ : Type u_1\nι : Type u_3\ninst✝¹ : Nonempty ι\nτ : Ω → WithTop ι\nE : Type u_4\nu : ι → Ω → E\ninst✝ : AddCommMonoid E\ns : Finset ι\ny : Ω\nω i : ι\nhi : ↑i = τ y\nhbdd : ↑i ∈ WithTop.some '' ↑s\nh : ω = (↑i).untopA\n⊢ (↑i).untopA ∈ s ∧ ↑i = ↑(↑i).untopA",
"ppTerm": "?mpr",
"assign... | [
"case mpr\nΩ : Type u_1\nι : Type u_3\ninst✝¹ : Nonempty ι\nτ : Ω → WithTop ι\nE : Type u_4\nu : ι → Ω → E\ninst✝ : AddCommMonoid E\ns : Finset ι\ny : Ω\nω i : ι\nhi : ↑i = τ y\nhbdd : ↑i ∈ WithTop.some '' ↑s\nh : ω = (↑i).untopA\n⊢ i ∈ s"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Probability.Moments.IntegrableExpMul | {
"line": 179,
"column": 4
} | {
"line": 179,
"column": 15
} | {
"line": 179,
"column": 16
} | [
{
"pp": "case refine_3\nΩ : Type u_1\nm : MeasurableSpace Ω\nX : Ω → ℝ\nμ : Measure Ω\nt : ℝ\nht_int_pos : Integrable (fun ω ↦ rexp (t * X ω)) μ\nht_int_neg : Integrable (fun ω ↦ rexp (-t * X ω)) μ\nh : Integrable (fun ω ↦ rexp (t * |X ω| + 0 * X ω)) μ\n⊢ Integrable (fun ω ↦ rexp (t * |X ω|)) μ",
"ppTerm": ... | [
"case refine_3\nΩ : Type u_1\nm : MeasurableSpace Ω\nX : Ω → ℝ\nμ : Measure Ω\nt : ℝ\nht_int_pos : Integrable (fun ω ↦ rexp (t * X ω)) μ\nht_int_neg : Integrable (fun ω ↦ rexp (-t * X ω)) μ\nh : Integrable (fun ω ↦ rexp (t * |X ω| + 0 * X ω)) μ\n⊢ Integrable (fun ω ↦ rexp (t * |X ω|)) μ"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
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